Optical network evaluation systems and methods

ABSTRACT

Embodiments of optical network evaluation systems and methods are disclosed. One system embodiment, among others, comprises a processor configured with logic to provide a cross layer model of disturbance propagation in an optical network based on a combination of a physical layer model and a network layer model, the physical layer mode based on the disturbance propagation having a threshold effect only on nodes along a route followed by the disturbance.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under agreementECS-0300305 awarded by the National Science Foundation. The Governmenthas certain rights in the invention.

CROSS-REFERENCE TO RELATED APPLICATION

This application is related to U.S. provisional patent applicationentitled “Cross-Layer Graphical Models for Resilience of All OpticalNetworks Under Cross-Talk Attacks,” filed on Jul. 26, 2005, and accordedSer. No. 60/702,481, which is entirely incorporated herein by reference.

TECHNICAL FIELD

The present disclosure is generally related to optical networks, and,more particularly, is related to systems and methods for evaluation ofoptical networks.

BACKGROUND

All-optical networks (AONs) are considered a promising technology fornext-generation optical networks. Major applications of AONs includemetropolitan area networks (MANs) and wide area networks (WANs), butMANs and WANs are not 100% secure. For instance, AONs are susceptible tomalicious (or unintentional) disturbances (e.g., attacks or other faultsthat propagate in a network) since the signals remain in the opticaldomain within the network, and are hence difficult to monitor closely.Further, due in part to the high data rates supported by AONs, evendisturbances of a short duration can result in a large amount of dataloss. Hence, security of AONs upon disturbances has become an importantissue, where an open question is how to incorporate security againstdisturbances in the design and engineering of AON architectures.Investigations of this question are important as AONs are still at anearly stage of implementation and ground-up developments of secureall-optical networks are possible.

There have been some approaches to this question in the past in thecontext of crosstalk attacks in AONs. Crosstalk in AONs can be caused bysignal leakage among different inputs at non-ideal network devices (e.g.optical switches), as illustrated in FIG. 1. FIG. 1 is a schematicdiagram of an optical network node 10 that illustrates crosstalkattacks, and in particular a detrimental type of crosstalk oftenreferred to as in-band crosstalk (e.g., where the crosstalk element iswithin the same wavelength as the signal). In-band crosstalk attacks canhappen at fiber links or network nodes. The optical network node 10comprises optical fibers 12, 14 coupled to demultiplexers 16 and 18. Thedemultiplexers 16 and 18 may be used in cooperation with optical filters(not shown). The optical network node 10 further comprises opticalswitches 20 and 22, which are coupled to demultiplexers 16, 18 byconnections 24 a, 26 a, 28 a, and 30 a. The optical switches 20 and 22are also coupled to multiplexers 32 and 34, via connections 24 b, 26 b,28 b, and 30 b, where signals are combined, and then output viaconnections 36 and 38. The demultiplexers 16 and 18 split the opticalsignals received on connections 12 and 14 into a plurality of bands ofdifferent wavelengths. For instance, signals of a first wavelength(e.g., λ₁) are provided on connections 24 a and 28 a, and signals of asecond wavelength (e.g., λ₂) are provided on connections 26 a and 30 a.As one exemplary mechanism of attack propagation, an attacker may gainlegitimate access to a network node at connection 24 a and insert asignal flow with strong signal power into the network. Due to thecrosstalk effects of wavelength switches, a small fraction of the signalfrom the attack channel (on connection 24 a) may leak into other normalchannels (e.g., connection 24 b, the leak graphically represented with adashed line) in a shared switching plane. The leakage superimposed ontonormal channels may exceed a predetermined threshold for a quality ofservice requirement, such that those channels are considered to beaffected by the attack at network nodes. In other words, AONs aresusceptible to crosstalk attacks.

As AONs grow in span and functionality, they have the potential toprovide services to a wider set of applications in the future (e.g.analog services, novel applications that require optical interfaces,etc.). Therefore, there is an increasing demand for access of the AONsfrom outside parties, such as limited management access to the networkfrom partners and customers of service providers, which results in anincreasing threat to optical network security. A wider set of users andan increasingly open platform of optical networks entail a higher riskof misuse of the network, which is evidenced by the security threatssuch as denial-of-service attacks and worm attacks in the currentInternet.

There have been several research activities with an aim to mitigate thethreats of crosstalk attacks in AONs, including attack detection basedon node wrappers, determination of necessary and sufficient conditionsfor crosstalk attack localization, and general frameworks for managingfaults and alarms in AONs. However, these approaches are reactive innature. Furthermore, certain crosstalk attacks are difficult to detect.For instance, sporadic crosstalk attacks may disrupt service but“disappear” before it can be detected.

SUMMARY

Embodiments of the present disclosure provide system and methods forevaluating optical networks.

One method embodiment, among others, comprises, for a physical model,modeling propagation of a disturbance in an optical network under staticnetwork conditions based on the disturbance propagation having athreshold effect only on nodes along a route followed by thedisturbance; for a network model, modeling a status of each networkroute in the optical network based on the disturbance; and combining thephysical layer model and the network layer model to provide a crosslayer model that characterizes the disturbance propagation based onnetwork layer and physical layer dependencies and interactions in theoptical network.

One system embodiment, among others, comprises a processor configuredwith logic to provide a cross layer model of disturbance propagation inan optical network based on a combination of a physical layer model anda network layer model.

Other systems, methods, features, and advantages of the presentdisclosure will be or become apparent to one with skill in the art uponexamination of the following drawings and detailed description. It isintended that all such additional systems, methods, features, andadvantages be included within this description, be within the scope ofthe present disclosure, and be protected by the accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the disclosure can be better understood with referenceto the following drawings. The components in the drawings are notnecessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 is a schematic diagram that illustrates exemplary crosstalkattack propagation in an optical network.

FIG. 2 is a schematic diagram of an example implementation for certainembodiments of optical network evaluation systems and methods.

FIG. 3 is a block diagram of an embodiment of the optical networkevaluation system shown in FIG. 2.

FIG. 4 is schematic diagram that illustrates an exemplary mesh networkwith a plurality of routes.

FIG. 5 is a schematic diagram that illustrates a directed probabilisticgraph representation of attack propagation for the mesh network shown inFIG. 4.

FIG. 6 is a schematic diagram that illustrates an undirectedprobabilistic graph representation of network routes for the meshnetwork shown in FIG. 4.

FIG. 7 is a schematic diagram that illustrates a factor graphrepresentation of the mesh network shown in FIG. 4.

FIGS. 8A-8C are schematic diagrams that illustrate exemplary ring,double-ring, and mesh networks.

FIG. 9 is a graph diagram that illustrates average network resilienceloss versus network load for the three networks shown in FIGS. 8A-8C.

FIG. 10 is a graph diagram that illustrates an average networkresilience loss versus network load for a National Science Foundation(NSF) benchmark network topology.

FIG. 11 is a flow diagram that illustrates an embodiment of an opticalnetwork evaluation method.

DETAILED DESCRIPTION

Disclosed herein are various embodiments of optical network (ON)evaluation systems and methods. In one embodiment, an ON evaluationsystem comprises a computer system configured to receive parameters of aparticular network and of a disturbance or attack, and responsivelygenerate physical layer and network layer models that are combined toprovide insight into disturbance propagation in the context ofinteractions and dependencies between the physical and network layers ofan optical network. Additionally, such an ON evaluation system can beused to determine the resiliency of optical networks to suchdisturbances, and hence provides a mechanism to address the resiliencyof optical networks to deliberate (e.g., attacks) or unintentionaldisturbances before such disturbances are detected and eliminated fromthe network.

Certain embodiments of ON evaluation systems and methods are describedbelow in the context of all optical networks (AONs), with theunderstanding that included within the scope of such networks are hybrid(electrical and optical) networks where evaluation is focused on theoptical portions of the hybrid network. Additionally, the variousembodiments of the ON evaluation systems are described herein in thecontext of disturbances embodied as crosstalk, with the understandingthat other disturbances, man-made, machine made, and/or inherent in thephysical architecture of the particular network, also apply and henceare considered within the scope of the disclosure.

Thus, the disclosure that follows describes the application ofprobabilistic graphical models to characterize cross-layer attackpropagation, including directed probabilistic graphs (e.g., BayesianBelief Networks) and undirected probabilistic graphs (e.g., MarkovRandom Fields). In particular, at the physical layer, a directedprobabilistic graph is described that models attack propagation understatic network traffic and a given source of attack with randomattacking power. At the network layer, an undirected probabilistic graphis described that represents the probability distribution of activeconnections. The ON evaluation systems and methods generate thesegraphical representations of the physical and network layer modelsthrough the execution of the various formulas and methods describedherein, and combines the physical- and the network-layer models togetherto form a cross-layer model that has a factor graph representation.

The cross-layer model is developed using a bottom-up approach andprovides an explicit representation of the complex dependencies betweenthe physical- and the network-layer. Furthermore, the graphical modelsfacilitate the analysis of multiple factors from network architecture onnetwork resilience. For regular topologies, bounds may be derived on thenetwork resilience. For irregular and/or large topologies, thecross-layer model provides computationally efficient methods forstudying the resilience where the analysis is not feasible.

In the description that follows, an exemplary implementation for certainembodiments of the ON evaluation systems and methods is provided,followed by a system embodiment. The remainder of the disclosure isorganized to illustrate an attack propagation model under static trafficand a given source of attack based on a directed probabilistic graph, anetwork-layer representation using an undirected probabilistic graph,and a cross-layer model based on a factor graph, and using thecross-layer model to quantify network resilience upon crosstalk attacks.Further illustration of an application of the models to the evaluationof network resilience of different network topologies is provided.

FIG. 2 is a schematic diagram that depicts a general purpose computer100 that serves as an example implementation for ON evaluation software,the latter represented with reference numeral 200. In one embodiment,the ON evaluation system comprises the computer 100. The general purposecomputer 100 can be in a stand-alone configuration, or networked amongother computers. The general purpose computer 100 includes a displayterminal 102 that provides a display of, among other things, arepresentation of the various models generated by the ON evaluationsoftware 200, such as the factor graph 700 shown in the display terminal102. Other screens may be presented, such as network topologies, userinterfaces for enabling user input of optical network parameters, etc.Although the factor graph 700 is shown on the display terminal 102,suggesting graphical representations of the various models generated bythe ON evaluation software 200, it should be understood by those havingordinary skill in the art that some embodiments of the ON evaluationsystem can be implemented in a manner that is transparent, in whole orin part, to the user. For instance, the basis of the various models arein some embodiments equations executed by the ON evaluation software200, and hence the results of such equation execution may be data thatis delivered to other software or devices without the need for agraphical representation.

The ON evaluation system can be implemented in software (e.g.,firmware), hardware, or a combination of the same. In the embodimentshown in FIG. 2, the ON evaluation system is embodied as a computer 100running an executable program (e.g., ON evaluation software 200), andthe program is executed by the general purpose computer 100 or otherspecial or general purpose digital computer, such as a personal computer(PC; IBM-compatible, Apple-compatible, or otherwise), workstation,minicomputer, or mainframe computer.

FIG. 3 is a block diagram showing a configuration of the general purposecomputer 100 that can implement the ON evaluation software 200.Generally, in terms of hardware architecture, the computer 100 includesa processor 302, memory 304, and one or more input and/or output (I/O)devices 306 (or peripherals) that are communicatively coupled via alocal interface 308. The local interface 308 can be, for example but notlimited to, one or more buses or other wired or wireless connections, asis known in the art. The local interface 308 may have additionalelements, which are omitted for simplicity, such as controllers, buffers(caches), drivers, repeaters, and receivers, to enable communications.Further, the local interface 308 may include address, control, and/ordata connections to enable appropriate communications among theaforementioned components.

The processor 302 is a hardware device for executing software,particularly that which is stored in memory 304. The processor 302 canbe any custom made or commercially available processor, a centralprocessing unit (CPU), an auxiliary processor among several processorsassociated with the computer 100, a semiconductor-based microprocessor(in the form of a microchip or chip set), a macroprocessor, or generallyany device for executing software instructions.

The memory 304 can include any one or combination of volatile memoryelements (e.g., random access memory (RAM, such as DRAM, SRAM, SDRAM,etc.)) and nonvolatile memory elements (e.g., ROM, hard drive, tape,CDROM, etc.). Moreover, the memory 304 may incorporate electronic,magnetic, optical, and/or other types of storage media. Note that thememory 304 can have a distributed architecture, where various componentsare situated remote from one another, but can be accessed by theprocessor 302.

The software in memory 304 may include one or more separate programs,each of which comprises an ordered listing of executable instructionsfor implementing logical functions. In the example of FIG. 3, thesoftware in the memory 304 includes the ON evaluation software 200,optional network topology software 310, and a suitable operating system(O/S) 312. Such software embodied in a computer readable medium such asmemory 304 collectively comprises logic in some embodiments, though notlimited to software stored in memory. The network topology software 310may comprise topology analysis software that can determine the amount ofnodes in a network and other operating parameters, and which can providesuch network parameters to the ON evaluation software 200 for use in thecomputation of equations corresponding to physical, network, and crosslayer models. The operating system 312 essentially controls theexecution of other computer programs, such as the ON evaluation software200 and the network topology software 310, and provides scheduling,input-output control, file and data management, memory management, andcommunication control and related services.

The ON evaluation software 200 is a source program, executable program(object code), script, or any other entity comprising a set ofinstructions to be performed. The ON evaluation software 200 can beimplemented, in one embodiment, as a distributed network of modules,where one or more of the modules can be accessed by one or moreapplications or programs or components thereof. In some embodiments, theON evaluation software 200 can be implemented as a single module withall of the functionality of the aforementioned modules. When a sourceprogram, the program is translated via a compiler, assembler,interpreter, or the like, which may or may not be included within thememory 304, so as to operate properly in connection with the O/S 312.

The I/O devices 306 may include input devices, for example but notlimited to, a keyboard, mouse, scanner, microphone, etc. Furthermore,the I/O devices 306 may also include output devices, for example but notlimited to, a printer, display, etc. Finally, the I/O devices 306 mayfurther include devices that communicate both inputs and outputs, forinstance but not limited to, a modulator/demodulator (modem foraccessing another device, system, or network), a radio frequency (RF) orother transceiver, a telephonic interface, a bridge, a router, etc.

When the computer 100 is in operation, the processor 302 is configuredto execute software stored within the memory 304, to communicate data toand from the memory 304, and to generally control operations of thecomputer 100 pursuant to the software. The ON evaluation software 200,the network topology software 310, and the O/S 312, in whole or in part,but typically the latter, are read by the processor 302, perhapsbuffered within the processor 302, and then executed.

When the ON evaluation software 200 is implemented in software, as isshown in FIG. 3, it should be noted that the ON evaluation software 200can be stored on any computer readable medium for use by or inconnection with any computer related system or method. The ON evaluationsoftware 200 can be embodied in any computer-readable medium for use byor in connection with an instruction execution system, apparatus, ordevice, such as a computer-based system, processor-containing system, orother system that can fetch the instructions from the instructionexecution system, apparatus, or device and execute the instructions.

In an alternative embodiment, where functionality of the ON evaluationsoftware 200 is implemented in hardware (the hardware providing the ONevaluation system functionality is also referred to herein as logic),such functionality can be implemented, in whole or in part, with any ora combination of the following technologies, which are each well knownin the art: a discrete logic circuit(s) having logic gates forimplementing logic functions upon data signals, an application specificintegrated circuit (ASIC) having appropriate combinational logic gates,a programmable gate array(s) (PGA), a field programmable gate array(FPGA), etc., or can be implemented with other technologies now known orlater developed.

Having described certain embodiments of an ON evaluation system, thefollowing description in association with FIGS. 4-7 explain anunderlying process employed by, and/or serving as a basis for, the ONevaluation software 200 to generate a physical layer model, a networklayer model, and a cross-layer model based on the combination of thephysical and network layer models. In some embodiments, functionality ofthe ON evaluation software 200 may be distributed among several moduleslocally or remotely located with respect to each other, wherein onemodule receives the physical and network layer models from anothermodule(s) co-located or remotely located (e.g., resident on one or moreexternal devices), and wherein the one module responsively generates thecross-layer model (and respective graphical representation found in thefactor graph 700). In particular, the description and accompanyingdrawings that follow generally focus on three, though not limited tothree, components of network architectures against crosstalk attacks:(a) physical layer optical devices, (b) physical topology, and (c)wavelength usage at the network layer, which is determined by networklayer traffic.

One goal of the various methods described herein is to quantify theeffects of these factors against crosstalk attacks, and hencecharacterize the interactions of at least these three factors of networkarchitectures during crosstalk attack propagation. For instance, attackspropagate to active wavelength channels of the same wavelength as theattacker's flow. Reference herein to a channel refers to frequency bandof one wavelength, although it should be understood in the context ofthis disclosure that multiple wavelength crosstalk attack (or otherdisturbance) propagation is considered to be within the scope of thisdisclosure. Meanwhile, wavelength usage at the network layer isdependent because of the sharing of network links among differentconnections. Therefore, the various models (whether representedgraphically or in equation form) provide an explicit representation ofthe cross-layer interactions.

Before proceeding with an explanation of FIGS. 4-7, a formulation of theproblem of solving attack propagation is presented below. The topologyof an AON is defined as an undirected graph G(V,E), with V being a setof nodes and E being a set of bi-directional links. Denote V_(i)˜V_(j)if there is one bi-directional link between V_(i) and V_(j), V_(i),V_(j)∈V. Let R be a finite set of routes in the network. Assume thatthere are no wavelength converters in the AON. A connection on router,r∈R, is defined as a bi-directional light-path on route r that consistsof one unidirectional flow in each direction. Each bi-directional linkdescribed below consists of two optical fibers, one for each direction.Hence, reference to a “connection” hereinafter includes a reference tobi-directional traffic, and reference to “flows” refers touni-directional traffic. Further, an assumption is made that eachwavelength can only be used by one active connection on the same networklink. Additionally, single-source in-band crosstalk attacks areconsidered in the examples that follow. That is, a crosstalk attack isstarted at the source node of a unidirectional flow on wavelength λ, andpropagates to flows that use the same wavelength. As the disclosure thatfollows focuses on in-band crosstalk attacks, “flows”, “connections”,and “channels” are typically used hereinafter without explicit referenceto their associations with wavelength λ.

Let S_(i) be a random variable that denotes the number of activechannels affected by the in-band crosstalk attack at the switching planeof node V_(i). Vector S=(S_(i):V_(i) ∈V) corresponds to the number ofaffected channels at each node in the network. Let N_(ij) denote thestatus of route r_(ij), where N_(ij)=1 if there is an active connectionon route r_(ij) between node V_(i) and V_(j), for r_(ij)∈R; N_(ij)=0,otherwise. Vector N=(N_(ij):r_(ij)∈R) then represents the status of allnetwork routes in R. Let f_(sd) be denoted as the flow starting fromnode s and terminating at node d. Given this problem formulation, it isunderstood in the context of this disclosure that one goal is to obtainthe following quantities to characterize attack propagation.

(a) P(S|N=n, R_(f)=f_(sd)): The probability of the number of channelsaffected at each network node given the status of network routes n andthe source of attack R_(f), where R_(f) denotes the unidirectional flowwhere the attack originates. This probability represents attackpropagation under a given n and f_(sd), and is to be characterizedthrough a directed probabilistic graph.

(b) P(N|R_(f)=f_(sd)): The probability of the status of network routesgiven the source of attack, which is to be described using an undirectedprobabilistic graph.

(c) P(S|R_(f)=f_(sd)): The probability of the number of channelsaffected at each node given the source of attack, which models attackpropagation under dynamic traffic. This probability combines thephysical- and the network-layer models from (a) and (b), and isdescribed with a factor graph representation.

The cross-layer model is then used to study network resilience based onthe resilience loss for a given attack flow and the average resilienceloss over all possible attack flows. Given that there is a crosstalkattack started on flow f_(sd), the network resilience loss can bedefined as follows:

$\begin{matrix}{{M_{f_{sd}} = {\sum\limits_{V_{i} \in V}{E_{f_{sd}}\left\lbrack S_{i} \right\rbrack}}},{{{where}\mspace{14mu}{E_{f_{sd}}\left\lbrack S_{i} \right\rbrack}} = {\sum\limits_{S_{i}}{s_{i}{P\left( {S_{i} = {{s_{i}\text{❘}R_{f}} = f_{sd}}} \right)}}}}} & (1)\end{matrix}$is the expected number of affected channels at node V_(i) given thesource of the attack. M_(f) _(sd) denotes the total number of activechannels affected when the attack starts from a particular flow.

Additionally, the average network resilience loss of the network can bedefined as M=E_(R) _(f) [M_(f) _(sd) ], where E_(R) _(f) [ ] stands forthe expectation over the source of the attack R_(f), i.e.,

$\begin{matrix}{{M = {\sum\limits_{f_{sd}}{M_{f_{sd}}{P\left( {R_{f} = f_{sd}} \right)}}}},{where}} & (2) \\{{{P\left( {R_{f} = f_{sd}} \right)} = {\frac{1}{2{R}}{P\left( {N_{sd} = 1} \right)}}},} & (3)\end{matrix}$with the assumption that each network route in R is equally likely to bean attacker's route, and the attack is started on one of the twounidirectional flows on the attacker's route with an equal probability.

Given the above-described formulation of the problem, reference is nowmade to FIG. 1 to illustrate the nature of an exemplary source of across-talk attack and how such an attack can propagate through anoptical network. The physical layer model is determined to model attackpropagation under static network traffic and a given source of attack.For example, one exemplary mechanism involves in-band crosstalk attackswhere an attacker gains legitimate access to the network and injectssignals of high power into a flow. Due to imperfect switching arrays,the attacker's channel (e.g., on connection 24 a) may affect otherchannels (e.g., on connection 24 b) that share the switching plane(e.g., the plane that supports the actual switching fabric for the inputsignals, such as switch 20 in FIG. 1), causing malfunctions at severallocations in the network. As explained above, FIG. 1 depicts an exampleof in-band crosstalk attack. At each network node, channels of the samewavelength (e.g., on connections 24 a, 28 a) from different input fibers12 and 14 share the same switching plane. Suppose that the crosstalk isinitiated on flow C1 (on connections 24 a) using wavelength λ₁ frominput fiber 12. All the wavelength channels that share a switching planewith C1 (e.g., channel C2 on connection 28 a from input fiber 14) may becontaminated by C1's power leakage.

In particular, a network node, such as network node 10 of FIG. 1, isaffected by the attack if the amount of in-band crosstalk incurred bynormal channels (e.g., channels in the network that are not theattacker's channel) at the switching plane of that node exceeds apredetermined threshold. Each node along the attacker's route may beaffected by the attack due to the high signal power of the attack flow,but the chance for nodes that are not on the attacker's route to beaffected by the attack is negligible. That is, normal flows affected bythe attack flow at one or more network nodes along the attacker's routedo not have attacking capability, as its signal power is unlikely to beincreased by more than half the normal channel power. For instance,consider the example in FIG. 1. Suppose, at one time instant, theattacker's jamming power is 20 dB higher than the normal channel powerand the optical switches have a crosstalk ratio of −35 dB. Then thepower of flow C2 is increased by around −15 dB of the normal channelpower at node 10. The power of flow C2 at a second node (not shown)coupled to optical fiber 38, for instance, is in the same order as innode 10, whose crosstalk leakage to flow leaving the second node isnegligible given the crosstalk ratio of −35 dB.

Currently, optical switches with crosstalk ratios much less than −35 dBare commercially available. Thus, in the models of the presentdisclosure, the in-band crosstalk caused by normal flows is ignored, andit is assumed that only nodes along the attacker's route may be affectedby the attack. Note that work prior to this disclosure on cross-talkattacks assumes that attacks may propagate to nodes that are not on theattacker's routes. Such a distinction from prior work renders theapproach described in this disclosure less complex and more informativethan prior approaches, as explained further below in association withFIG. 5. In addition, attacks propagate to all the active channels thatshare the switching plane with the attacker's channel at each affectednode. Based on these assumptions, the probabilistic attack propagationmodel can be described in the following sections.

A physical layer model is derived based on the following considerations.Consider a crosstalk attack started at node s on flow f_(sd). Let theset of nodes traversed by flow f_(sd) be V_(f) _(sd) ={V₁, V₂, . . . ,V_(k)}, where V₁ and V_(k) are the source and the destination nodesrespectively. The attack propagation is characterized by the status ofeach node in V_(f) _(sd) and the status of wavelength channels at theswitches of those nodes.

The status of node V_(i) can be defined as a binary variable X_(i).Specifically, let the signal power of a normal flow at the switchingplane of each node be u_(n) when there is no attack in the network. Letthe crosstalk ratio of the switches in the network be l_(c) and letc_(th) be a predetermined constant. Then X_(i)=1 if the amount ofin-band crosstalk incurred by a normal channel at the switching plane ofnode V_(i) exceeds c_(th)u_(n); X_(i)=0, otherwise. Furthermore, nodeV_(i) is affected by the attack if X_(i)=1.

The use of binary nodal states facilitates the investigation ofcrosstalk attack propagation with random attacking power at the source.To be specific, the amount of in-band crosstalk at each node underattack may have a wide range of values. But the binary status ofcrosstalk levels at a node is simple and often observable. In fact, afrequent scenario for attack detection and monitoring is whether apredetermined threshold or service guarantee is violated at each node.When the amount of crosstalk is below the threshold, the node is “up”,i.e., operational; otherwise, the node is “down”, i.e., affected.

Hence the attacker's jamming power is treated as a random variable thatobeys a certain probability distribution. The status of network nodesunder crosstalk attacks then becomes (i.e., is represented as) binaryrandom variables. The randomness lies in the fact that the crosstalklevel is random due to the random jamming power of the attack. If theattacker's jamming power has a higher probability of being large, it ismore likely for the attack to propagate farther away from the sourcenode.

To determine the status of each node under attack, consider theattenuation of the jamming power of flow f_(sd) along its route. Thesignal power of flow f_(sd) in the switching plane of node V_(i) isdenoted as a random variable U_(i), i=1, 2, . . . k. The attenuation ofthe jamming power of f_(sd) along its route can be captured usingdeterministic composite functions that depend on the characteristics ofoptical devices. Additionally, it is assumed that there exists an inputerbium-doped fiber amplifier (EDFA) and an output EDFA at each side of anode (e.g., node 10 of FIG. 1), respectively. Note that in someembodiments, the techniques described herein can be extended to thederivation of physical layer models using other amplifiers (e.g., Ramanamplifiers, etc.). Furthermore, the following parameters can be defined:

l_(i,1): Signal loss ratio of node V_(i) before the flow enters theswitching plane, which mainly includes signal loss at a demultiplexer(e.g., demultiplexers 16 and 18, as shown in FIG. 1).

l_(i,2): Signal loss ratio of node V_(i) after the flow enters theswitching plane, which mainly includes loss at the switching plane and amultiplexer (e.g., multiplexers 32 and 34).

a_(i,j): Signal loss ratio of the fiber span between node V_(i) and nodeV_(j).

g_(i,1)( ): The gain of the EDFA at the input side of node V_(i).

g_(i,2)( ): The gain of the EDFA at the output side of node V_(i).

For a given network, l_(i,1), l_(i,2), a_(i,j) are constants; g_(i,1)( )and g_(i,2)( ) are deterministic non-linear functions of the input powerto the amplifiers (e.g., EDFAs).

Additionally, the following gain model for EDFAs can be adopted:

$\begin{matrix}{{g_{ij}\left( P_{input} \right)} = \left\{ \begin{matrix}{d_{ij},} & {{{{if}\mspace{14mu} P_{input}} \leq p_{th}},} \\{{1 + {\frac{p_{sat}}{P_{input}}\log\;\frac{g_{0}}{g_{ij}\left( P_{input} \right)}}},} & {{otherwise},}\end{matrix} \right.} & (4)\end{matrix}$where P_(input) is the total input power; p_(sat) is the internalsaturation power; g₀ is the small signal saturated gain; p_(th) is theinput power threshold for successful gain clamping, and d_(ij) is theclamped gain value.

Assume that the attacker's flow (f_(sd)) does not share EDFAs with otherflows. This corresponds to a conservative model of the jamming powerattenuation and a worst-case scenario of in-band attack propagation, asall the photons of the EDFAs are used to amplify the attacker's signal.Then,U _(i+1) =l _(i+1,1)π_(i+1,1)(a _(i,i+1)π_(i,2)(l _(i,2) U _(i))),  (5)where π_(i,j)=P_(input)g_(i,j)(P_(input)) is the output power of theEDFA with gain g_(ij)(P_(input)) and input power P_(input). Then,composite function τ_(j−1,j)(τ_(j−2,j−1)( . . . τ_(i,j+1)(.))) capturesthe attenuation of the jamming power between node V_(i) and V_(j).

Further, assume that, when there is no crosstalk attack in the network,amplifiers on each fiber operate in the gain clamped regions and make upthe signal attenuation between two nodes. Furthermore, assume that theattacker's jamming power at the source node of the attack follows acumulative distribution function η(U) with minimum power u_(min),u_(min)≧c_(th)u_(n)/l_(c), and maximum power u_(max). Then, it can beshown that the status of each node along the attacker's route, X_(i),i=1, 2, . . . , k, form a Markov Chain. Specifically,

$\begin{matrix}{{P\left( {X_{1} = 1} \right)} = 1.} & (6) \\{{{P\left( {{X_{i + 1}\text{❘}X_{1}},X_{2},\ldots\mspace{11mu},X_{i}} \right)} = {P\left( {X_{i + 1}\text{❘}X_{i}} \right)}},{i = 1},2,\ldots\mspace{11mu},{k - 1.}} & (7) \\{{P\left( {X_{i + 1} = {{1\text{❘}X_{i}} = 0}} \right)} = 0.} & (8) \\{{{P\left( {X_{i + 1} = {{1\text{❘}X_{i}} = 1}} \right)} = {\frac{P\left( {U_{i + 1} > {c_{th}/I_{c}}} \right)}{P\left( {U_{i} > {c_{th}/I_{c}}} \right)} = \frac{1 - {\eta\left( \delta_{1,{i + 1}} \right)}}{1 - {\eta\left( \delta_{1,i} \right)}}}},} & (9)\end{matrix}$where δ_(1,i), 1≦i≦k−1, corresponds to the minimum value of jammingpower at node V₁ such that attack can propagate to node V_(i), andsatisfies,τ_(i−1,i)(τ_(i−2,i−1)( . . . τ_(1,2)(δ_(1,i))))=c_(th)u_(n)/l_(c).  (10)

The derivation of (7) to (9) can be explained as follows:

Let U_(i) denote the jamming power of the attack flow at node V_(i),∀V_(i)∈V_(f) _(sd) . It can be shown that if U_(i)≧U_(i+1) for 1≦i<k,then P(X_(i+1)|X₁, X₂ . . . , X_(i))=P(X_(i+1)|X_(i)). SupposeU_(i)≧U_(i+1). Since X_(i)=1, if U_(i)>c_(th)/(l_(c)u_(n)); and X_(i)=0,otherwise. It follows that P(X₁=x₁, X₂=x₂, . . . , X_(i)=x_(i))≠0, onlyif x₁≧x₂≧ . . . ≧x_(i). Let k₁=max{j:x_(j)=1 & 1≦j≦i}, which is thelargest index of nodes affected by the attack among V₁, V₂, . . . ,V_(i). Then,(a) If 1≦k₁<i, X₁=1, . . . X_(k) ₁ =1, X_(k) ₁ ₁=0, . . . , X_(i)=0.Therefore

$\begin{matrix}\begin{matrix}{{P\begin{pmatrix}{{X_{i + 1} = {{0❘X_{1}} = 1}},{{\ldots\mspace{11mu} X_{k_{1}}} = 1},} \\{{X_{k_{1} + 1} = 0},\ldots\mspace{11mu},{X_{i} = 0}}\end{pmatrix}} = \frac{P\begin{pmatrix}{{X_{1} = 1},\ldots\mspace{11mu},{X_{k_{1}} = 1},} \\{{X_{k_{1} + 1} = 0},\ldots\mspace{11mu},{X_{i} = 0},} \\{X_{i + 1} = 0}\end{pmatrix}}{P\begin{pmatrix}{{X_{1} = 1},\ldots\mspace{11mu},{X_{k_{1}} = 1},} \\{{X_{k_{1} + 1} = 0},\ldots\mspace{11mu},{X_{i} = 0}}\end{pmatrix}}} \\{= {\frac{P\left( {{X_{k_{1}} = 1},{X_{k_{1} + 1} = 0}} \right)}{P\left( {{X_{k_{1}} = 1},{X_{k_{1} + 1} = 0}} \right)} = 1.}}\end{matrix} & (11)\end{matrix}$Since P(X_(i−1)=0|X_(i)=0)=1, the following is observed:P(X _(i−1) |X ₁ ,X ₂ . . . , X _(i))=P(X _(i+1) |X _(i)).(b) If 1≦k₁=i, X₁=1, . . . , X₁=1. Therefore,

$\begin{matrix}\begin{matrix}{{P\left( {{X_{i + 1} = {{0\text{❘}X_{1}} = 1}},\ldots\mspace{11mu},{X_{i} = 1}} \right)} = \frac{P\left( {{X_{1} = 1},\ldots\mspace{11mu},{X_{i} = 1},{X_{i + 1} = 0}} \right)}{P\left( {{X_{1} = 1},\ldots\mspace{11mu},{X_{i} = 1}} \right)}} \\{= \frac{P\begin{pmatrix}{{U_{1} > {c_{th}/\left( {I_{c}u_{n}} \right)}},\ldots\mspace{11mu},{U_{i} >}} \\{{c_{th}/\left( {I_{c}u_{n}} \right)},{U_{i + 1} < {c_{th}/\left( {I_{c}u_{n}} \right)}}}\end{pmatrix}}{P\left( {{U_{1} > {c_{th}/\left( {I_{c}u_{n}} \right)}},\ldots\mspace{11mu},{U_{i} > {c_{th}/\left( {I_{c}u_{n}} \right)}}} \right)}} \\{= \frac{P\left( {{U_{i} > {c_{th}/\left( {I_{c}u_{n}} \right)}},{U_{i + 1} < {c_{th}/\left( {I_{c}u_{n}} \right)}}} \right)}{P\left( {U_{i} > {c_{th}/\left( {I_{c}u_{n}} \right)}} \right)}} \\{= \frac{P\left( {{X_{i} = 1},{X_{i + 1} = 0}} \right)}{P\left( {X_{i} = 1} \right)}} \\{= {{P\left( {X_{i + 1} = {{0\text{❘}X_{i}} = 1}} \right)}.}}\end{matrix} & (12)\end{matrix}$

Next, it can be shown that U_(i)≧U_(i−1), assuming that, when there isno crosstalk attack in the network, amplifiers on each fiber operate inthe gain clamped regions and make up the signal attenuation between thetwo nodes. From (5) above, the following is observed.U _(i+1)=τ_(i,i+1)(U _(i))=l _(i+1,1)π_(i+1,1)(a _(i,i+1)π_(i,2)(l_(i,2) U _(i))),andl _(i+1,1) a _(i,i+1) l _(i,2) d _(i,2) d _(i+1,1)=1,  (13)where d_(i,2) denotes the clamped gain of EDFA at the output side ofnode V_(i); d_(i+1,1) denotes the clamped gain of EDFA at the input sideof node V_(i+1). Then,If l_(i,2)U_(i)≦p_(th,(i,2)), U_(i+1)=U_(i),  (14)If l _(i,2) U _(i) >p _(th,(i,2)), π_(i,2)(l _(i,2) U _(i))<d _(i,2) U_(i),  (15)which corresponds to the case where the EDFA with subscript (i,2) worksat the saturation region. Therefore,U_(i+1)<l_(i+1,1)d_(i+1,1)a_(i,i+1)d_(i,2)l_(i,2)U_(i).

It follows that U_(i)≧U_(i+1). To prove (9) above, it suffices to showthat τ_(i,i+1) (U_(i)) monotonically increases in U_(i), whereU _(i+1)=τ_(i,i+1)(U _(i))=l _(i+1,1)π_(i+1,1)(a _(i,i+1)π_(i,2)(l_(i,2) U _(i))).

Since l_(i+1,1), a_(i,i+1), and l_(i,2) are constants, to show thatτ_(i,i+1) (U_(i)) monotonically increases in U_(i), it suffices to showthat π_(ij) (P_(input)) monotonically increase in P_(input). This can beobtained by showing

$\frac{\partial\left( {P_{input}{g\left( P_{input} \right)}} \right)}{\partial P_{input}} > {0\mspace{14mu}{for}\mspace{14mu}{(4).}}$This means that the higher the input power at EDFA, the higher theoutput power when the EDFA works at either the saturated or thenon-saturated region.

The conditional probabilities in (9) can take different forms dependingon η(U). For simplicity, the following can be denoted:

$\begin{matrix}{{{P\left( {X_{i + 1} = {{1\text{❘}X_{i}} = 1}} \right)} = \alpha_{i}},{{{where}\mspace{14mu}\alpha_{i}} = \frac{1 - {\eta\left( \delta_{1,{i + 1}} \right)}}{1 - {\eta\left( \delta_{1,i} \right)}}},} & (16)\end{matrix}$where δ_(1,i), 1≦i<k, as in (10) above.

If it is further assumed that the attacker's jamming power at the sourcenode of the attack is uniformly distributed in [u_(min),u_(max)], (9)can be rewritten as follows:

$\begin{matrix}{{P\left( {X_{i + 1} = {{1\text{❘}X_{i}} = 1}} \right)} = {\frac{\max\left\{ {0,{u_{\max} - {\max\left\{ {u_{\min},\delta_{1,{i + 1}}} \right\}}}} \right\}}{u_{\max} - {\max\left\{ {u_{\min},\delta_{1,i}} \right\}}}.}} & (17)\end{matrix}$

Hereinafter in this disclosure, it is assumed that α_(i)'s are known.

The number of active channels affected by the attack at the switchingplane of node V_(i) is considered. Let R_(ij) be the set of networkroutes that use link ij. Under static traffic,

$\sum\limits_{r_{uv} \in R_{ij}}n_{uv}$corresponds to the number of flows that enter the switching plane ofnode V_(i) through link ij;

$\sum\limits_{r_{ih} \in R_{ij}}n_{ih}$corresponds to the number of flows that are locally originated from nodeV_(i) and enter the network through link ij. Hence, under static networktraffic, the total number of affected channels at the switching plane ofV_(i), V_(i)∈V_(f) _(sd) , is given by the following equation:

$\begin{matrix}{{P\left( {{S_{i} = {{s_{i}❘X_{i}} = x_{i}}},{N = n},{R_{f} = f_{sd}}} \right)} = \left\{ \begin{matrix}{1,} & {{{{if}\mspace{14mu} s_{i}} = {{\sum\limits_{V_{i}\sim V_{j}}{\left\{ {{\sum\limits_{r_{uv} \in R_{ij}}n_{uv}} + {\sum\limits_{r_{ih} \in R_{ij}}n_{ih}}} \right\}\mspace{14mu}{and}\mspace{14mu} x_{i}}} = 1}},} \\1 & {{{{if}\mspace{14mu} s_{i}} = {{1\mspace{14mu}{and}\mspace{14mu} x_{i}} = 0}},} \\{0,} & {{otherwise}.}\end{matrix} \right.} & (18)\end{matrix}$

Equation (18) provides that, when node V_(i) is affected by the attack,all the active channels at the switching plane of V_(i) are affected bythe attack; otherwise, if node V_(i) is not affected by the crosstalkattack, only one channel (e.g., only the channel used by flow f_(sd)itself) is affected by the attack at the node. That is, equation (18)describes a probability distribution of the number of channels affectedby the attack given the following parameters: (a) the status of nodeX_(i); (b) the status of each network route in the network, n; and (c)the source of the attack flow f_(sd). In equation (18),

$\sum\limits_{r_{uv} \in R_{ij}}n_{uv}$corresponds to the number of flows that enter the switching plane ofnode V_(i) through link ij;

$\sum\limits_{r_{ih} \in R_{ij}}n_{ih}$corresponds to the number of flows that are locally originated from nodeV_(i) and enter the network through link ij. Therefore,

$\sum\limits_{V_{i}\sim V_{j}}\left\{ {{\sum\limits_{r_{uv} \in R_{ij}}n_{uv}} + {\sum\limits_{r_{ih} \in R_{ij}}n_{ih}}} \right\}$corresponds to the total number of active wavelength channels in theswitching plane of node V_(i). Hence, equation (18) describes that whennode V_(i) is affected by the attack, all the active channels at theswitching plane of V_(i) are affected by the attack; otherwise, if nodeV_(i) is not affected by the crosstalk attack, only one (e.g., only thechannel used flow f_(sd) itself), is affected by the attack at the node.

Combining equations (16) and (18) results in a physical-layer attackpropagation model, expressed by the following equation:

$\begin{matrix}{{{P\left( {X_{f_{sd}},{{s_{f_{sd}}\text{❘}N} = n},{R_{f} = f_{sd}}} \right)} = {\prod\limits_{i = 1}^{k - 1}{{P\left( {{X_{i + 1}\text{❘}X_{i}},{R_{f} = f_{sd}}} \right)}{\prod\limits_{i = 1}^{k}{P\left( {{S_{i}\text{❘}X_{i}},{N = n},{R_{f} = f_{sd}}} \right)}}}}},} & (19)\end{matrix}$where S_(f) _(sd) =(S_(i):V_(i) ∈V_(f) _(sd) ), X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ), which is the status of nodes in theattacker's route, and k is the number of node in V_(f) _(sd) .Therefore, under static network traffic (given N=n),(X_(i),S_(i):V_(i)∈V_(f) _(sd) ) forms a directed probabilistic graph(Bayesian Belief Network). Each node in the probabilistic graphrepresents either X_(i) or S_(i). There is one directed edge from X_(i)to X_(i+1) and one directed edge from X_(i) to S_(i) respectively. Notethat, given N=n and X_(i)=x_(i), S_(i) is deterministic, but S_(i) isincluded for an explicit graphical representation of attack propagation.Note that equation (18) provides a detailed form of a conditionalprobability in equation (19), and equations (6) through (9), (16) and(17) correspond to a derivation of a second conditional probability inequation (19).

Having described the derivation of a physical layer model, reference isnow made to FIG. 4. FIG. 4 is a schematic diagram that illustrates anexemplary mesh network 400 with a plurality of routes (e.g., elevenroutes). That is, the exemplary mesh network 400 comprises a pluralityof nodes 402 (seven nodes 402 are shown labeled A, B, C, D, E, F, and G)and a plurality of routes 404 (eleven routes, where all the routes in Rare marked in dashed lines). Suppose that the crosstalk attack isstarted on flow BD 406. Reference herein to an attacker's route (ordisturbance route) refers to the route traversed between an attacker'sflow. For instance, with reference to FIG. 4, if the attacker's flow isf_(bd), then the attacker's route is the route from node B to node D,which passes through links BC and CD. In certain embodiments, a node isconsidered as affected by an attack if the amount of in-band crosstalkincurred by normal channels from that particular node exceeds apredetermined threshold. Branches off the route BD (e.g., the branchfrom node B to node F) are not part of the attacker's route. Asexplained above, in prior work, nodes that are not part of theattacker's route are assumed to be affected by the attack, in contrastto the various embodiments described herein, as further described belowin conjunction with FIG. 5. The directed probabilistic graphrepresentation 500 of attack propagation is shown in FIG. 5.

Referring to the directed graph representation 500 of FIG. 5 and themesh network 400 of FIG. 4, a disturbance may propagate from node B tonode C, then further from node C to node D, as evidenced by thedirectional edge from node B to node C and from C to D. Whether a nodeis affected by the attack or not is denoted by the variables Xi, where iin this example equals B, C, and D. Meanwhile, the number of affectedchannels at each node can be found from variable(s) S. Under staticnetwork traffic, the number of affected channels at each node isdetermined by the status of that node, which is indicated by directededges from variable Xi to variable Si, where i equals B, C, and D.

Several differences are noted between the directed graph 500 of FIG. 5and previous work on cross-talk attack models. For instance, in theMarch 2005 publication authored by the inventors of the currentdisclosure and entitled, “Probabilistic Graphical Models for Resilienceof All-Optical Networks under Crosstalk Attacks,” it is assumed thatattack propagation includes nodes not located on the attacker's route,as explained above. Such assumptions at the time of that publicationwere based in part on the relative infancy in the field of security inoptical networks and less advanced optical technologies related tooptical switches and crosstalk ratios. Under such prior assumptions andreferring to FIG. 5, additional directed edges would be shown from X_(B)to neighboring nodes that can be affected by the attack, for instance anX_(A) located in FIG. 5 to the left of X_(B) or an X_(F) located belowX_(A), or similarly to a neighboring node below or beyond X_(D). Incontrast, and referring to FIG. 5, the physical layer model involves thestatus of nodes only on the attacker's route.

Referring to FIG. 5, and drawing further distinctions between thedisclosure and the March 2005 publication, X_(B) represents whether nodeB is affected by the attack or not. S_(B) represents the number ofaffected channels in the switching plane of node B. To contrast withprior work in this field, assume the S_(B) is replaced with an X_(F).According to the March 2005 publication authored by the inventors andreferenced above, a directed edge from X_(B) to X_(F) represents theconditional probability of node F (which is not along an attacker'sroute) being affected by the attack given the status of node B. That is,in the March 2005 publication, consideration is given of an attack thatpropagates from node B to node F. In contrast, the directed edge fromnode X_(B) to node S_(B) represents the conditional probability of thenumber of affected channels in the switching plane of node B given thestatus of node B. The current physical layer model considers the damagecaused by the attack at a particular node given its status.

Additionally, the March 2005 publication fails to teach or suggest howto determine the probability of attack (e.g., probability that an attackpropagates from node B to C and from C to D, etc.). That is, one skilledin the art is not able to derive the parameters of the models based onthe March 2005 disclosure. Further, and as explained above, previousapproaches lack the directed edge pertaining to the variable S, andhence fail to provide a graphical representation of attack propagation(e.g., the status of each node along an attacker's route).

Note that in context of modeling AON signal transmissions, thephysical-layer model described above is developed with the followingassumptions: (1) the in-band crosstalk due to channels with normalsignal power and/or nonlinear effects is ignored; (2) under normaloperations, the EDFAs work at a gain-clamped region and make up for thesignal losses between two network nodes; (3) the optical switches havethe same crosstalk ratio and threshold of crosstalk leakage for thedefinition of node affection. Note that reference to node affection orthe like refers to a node being “affected” by the attack. For instance,a node is affected by an attack if the amount of in-band crosstalkincurred by normal channels at the switching plane of that node exceedsa predetermined threshold for quality of service requirement. Assumption(1) is reasonable because of the low crosstalk ratio of current opticalswitches. If assumption (2) is relaxed (e.g., does not hold) so that theEDFAs work at gain-clamped region under normal operations, but may makeup for more than the signal losses between two network nodes, then thestatus of nodes along the attacker's route may still form a MarkovChain. However, the order of nodes in the Markov Chain does notnecessarily follow the sequence of X₁, X₂, . . . , X_(k). The same istrue, if assumption (3) is relaxed so that optical switches in thenetwork have different crosstalk ratios or the thresholds of crosstalkleakage for the definition of node affection are heterogeneous fordifferent nodes.

Another approach to model the attack propagation is to define a randomvariable that corresponds to the position of the last affected nodealong the attacker's route. This model is equivalent to the Markov Chainmodel in (6) to (16), but does not include explicitly the status of eachnode. Thus, such a model does not visually signify the actual attackpropagation along the attacker's route.

The physical layer model described by Equation (19) characterizes attackpropagation under static network traffic. Under dynamic traffic,however, the status of each network route N_(sd), r_(sd)∈R, is randomand can be characterized using a network layer model. To obtain thenetwork layer model, P(N|R_(f)=f_(sd)) is to be obtained, which is theprobability distribution of route status given the source of the attack.From (3) above, the following equation is provided:P(N|R _(f) =f _(sd))=P(N|N _(sd)=1).  (20)Then it suffices to find P(N), which can be characterized by anundirected probabilistic graph.

An undirected probabilistic graph can be represented as G=(V, E), whereV represents a set of vertices, and E represents a set of edges. Eachnode V_(i)∈V represents a random variable. A subset of nodes V_(C) issaid to separate two other subsets of nodes V_(A) and V_(B) if everypath joining every pair of nodes V_(i)∈V_(A) and V_(j)∈V_(B) has atleast one node from V_(C). An undirected probabilistic graph implies aset of conditional independence relations. That is, for any disjointsubsets of nodes in the undirected graph, V_(A), V_(B), and V_(C), ifV_(C) separates V_(A) and V_(B), then V_(A) and V_(B) are conditionalindependent given V_(C). A node is separated from other nodes in theundirected graph by all its neighbors.

A clique denotes a subset of V that contains either a single node orseveral nodes that are all neighbors of one another. Then the jointprobability distribution of V has a product form, as shown below:

$\begin{matrix}{{{P(V)} = {Z^{- 1}{\prod\limits_{q \in C}{\psi_{q}\left( \left\{ {{V_{i}\text{:}V_{i}} \in V_{q}} \right\} \right)}}}},} & (21)\end{matrix}$where Z is the normalizing constant,

${Z = {\sum\limits_{V}{\prod\limits_{q \in C}{\psi_{q}\left( \left\{ {{V_{i}\text{:}V_{i}} \in V_{q}} \right\} \right)}}}};\psi_{q}$is a non-negative function defined for clique V_(q)∈C, and C denotes theset of all the cliques in the graph G.

The network-layer model is formed as follows. Each vertex in theundirected probabilistic graph represents the status of a route N_(ij),r_(ij)∈R. Furthermore, the status of all network routes that share thesame link forms a clique. It has been shown that the steady statedistribution of the number of calls in loss networks without controlform a Markov Random Field (MRF), which is one type of undirectedprobabilistic graph. The MRF representation can be generalized to anundirected probabilistic graph representation that includes explicitlythe dependence among different routes due to the capacity constraint andthe network load.

For the exemplary mesh network 400 shown in FIG. 4, the correspondingnetwork-layer model 600 is shown in FIG. 6. Referring to FIGS. 4 and 6,consider route AC 408, which traverses two network links: AB and BC.Meanwhile, link AB is in route AB 410 and route AF 412; link BC is inroute BC 414 and route BD 406. Since wavelength λ can only be used byone connection on each network link, route AC 408 has a contention ofwavelength usage with routes AB 410, AF 412, BC 414, and BD 406.However, once the status of routes AB 410, AF 412, BC 414 and BD 406 isknown, the status of route AC 408 can be determined without violatingthe capacity constraints. Hence, routes AB 410, AF 412, BC 414 and BD406 are neighbors of route AC 408 and separate route AC from routes inthe rest of the network, as shown in FIG. 6.

Therefore, by defining routes that share the same network link asneighbors, the capacity constraint in the undirected probabilistic graphis captured. The probability distribution of all network routes can beobtained by specifying proper clique potentials based on (21). Theclique potentials are selected to characterize both the dependenciesamong different network routes and the varying network load.

In the discussion of physical layer models above, R_(ij) is denoted as asubset of routes in R that traverse link ij. A clique, denoted asC_(ij), can then be formed with all the routes in R_(ij). Then thepotential function of C_(ij), denoted as ψ_(ij), is obtained as follows:(1) ψ_(ij)≠0 if and only if the capacity constraint is satisfied (e.g.,at most one route in R_(ij) is active); (2) if the wavelength is used onlink ij, then ψ_(ij)=γ_(ij); otherwise, ψ_(ij)=1−γ_(ij), 0<γ_(ij)<1.From (21), the joint probability of all routes satisfies the followingequation:

$\begin{matrix}{{{{P(N)} = {\frac{1}{Z_{N}}{\prod\limits_{({V_{i}\sim V_{j}})}{{\gamma_{ij}^{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}\left( {1 - \gamma_{ij}} \right)}^{({1 - {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}})}{I_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)}}}}},\mspace{79mu}{{{{where}\mspace{14mu}{I_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)}} = {{1\mspace{14mu}{if}\mspace{14mu}{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}} = {0\mspace{14mu}{or}\mspace{14mu} 1}}};\mspace{14mu}{and}}}\mspace{79mu}{{{I_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)} = 0},{{otherwise}.}}} & (22)\end{matrix}$The clique functions are non-zero if and only if l₁(Σ_(r) _(uv) _(∈R)_(ij) N_(uv))=1. Thus (22) characterizes the dependencies of routes thatresult from the capacity constraints. For instance, the potentialfunction for the clique that corresponds to the routes using link AB,i.e., route N_(AB), N_(AF), and N_(AC), isψ_(AB)=γ_(AB) ^((N) ^(AB) ^(+N) ^(AF) ^(+N) ^(AC) ⁾(1−γ_(AB))^(1−(N)^(AB) ^(+N) ^(AF) ^(+N) ^(AC) ⁾l₁(N_(AB)+N_(AF)+N_(AC)).  (23)Note that (N_(AB)+N_(AF)+N_(AC)) corresponds to the total number ofactive connections using wavelength λ at link AB. As each wavelength canbe used by at most one active connection, (N_(AB)+N_(AF)+N_(AC))=0 or 1.Thus,

${I_{1}\left( {N_{AB} + N_{AF} + N_{AC}} \right)} = \left\{ \begin{matrix}1 & {{{{{if}\mspace{14mu} N_{AB}} + N_{AF} + N_{AC}} = {0\mspace{14mu}{or}\mspace{14mu} 1}},} \\0 & {{otherwise}.}\end{matrix} \right.$where γ_(AB) can be considered as the weight of using wavelength λ atlink AB. The potential function is γ_(AB) if wavelength λ is used atlink AB; and 1−γ_(AB) if wavelength λ is not used at link AB.

The above network layer model equation (22) represents an advancementover the network layer model described in previous work by theinventors, namely the March 2005 publication, “Probabilistic GraphicalModels for Resilience of All-Optical Networks under Crosstalk Attacks.”For instance, the network layer model described herein only involves thestatus of each route (represented by parameter, N), which is a simpleand straightforward formulation for the network layer model. Incontrast, the network layer model in the March 2005 publicationadditionally involves the status of each link (represented by theparameter, W), as follows:

${{P\left( {N,W} \right)} = {\frac{1}{Z_{N}}{\prod\limits_{({V_{i}\sim V_{j}})}{{\gamma_{ij}^{w_{ij}}\left( {1 - \gamma_{ij}} \right)}^{({1 - W_{ij}})}{I_{2}\left( {{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}},W_{ij}} \right)}}}}},{{{where}\mspace{14mu}{I_{2}\left( {{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}},W_{ij}} \right)}} = {{1\mspace{14mu}{if}\mspace{14mu}{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}} = {W_{ij}\mspace{14mu}{and}}}}$W_(ij) ∈ {0, 1}; and${{I_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)} = 0},{{otherwise}.}$

The network load (e.g., the probability that wavelength λ is used in thenetwork) is characterized by parameter, γ_(ij)·γ_(ij) can be consideredas a “weight” for using a wavelength at link ij; 1−γ_(ij) can beconsidered as a “weight” for not using a wavelength at link ij. Whenγ_(ij)≡γ, ∀V_(i)˜V_(j), γ can be related to the network load as follows:

Proposition 1: Let ρ denote the network load,

$\rho = {{E_{P{(N)}}\left\lbrack \frac{\sum\limits_{V_{i}\sim V_{j}}{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}}{E} \right\rbrack}.}$If in (22), γ_(ij)≡γ, ∀V_(i)˜V_(j), then ρ monotonically increases in γ.A detailed proof of Proposition 1 can be explained as follows:To prove Proposition 1, it suffices to show that

$\frac{\partial\rho}{\partial\gamma},{\forall{0 < \gamma < 1.}}$From (22), assume γ_(ij)≡γ, then,

$\begin{matrix}{{P(N)} = {\frac{1}{Z_{N}}{\prod_{({V_{i}\sim V_{j}})}{{\gamma^{\sum\limits_{r_{sd} \in R_{ij}}N_{sd}}\left( {1 - \gamma} \right)}^{({1 - {\sum\limits_{r_{sd} \in R_{ij}}N_{sd}}})}{{I_{1}\left( {\sum\limits_{r_{sd} \in R_{ij}}N_{sd}} \right)}.}}}}} & (24)\end{matrix}$

Let

$W_{ij} = {{\sum\limits_{r_{uv} \in R_{ij}}{N_{uv}\mspace{14mu}{and}\mspace{14mu} W}} = {\left( {W_{ij}\text{:}{\left. V_{i} \right.\sim V_{j}}} \right).}}$W is a vector that represents the wavelength usage at each link in thenetwork. A configuration of (N, W) with non-zero probability is denotedas a traffic pattern, i.e., a traffic pattern (N, W) satisfies thecapacity constraints and P(N=n, W=w)>0. Let T_(k), k=0, 1, . . . , |E|,be the set of traffic patterns that k links in the network are used byactive connections, with |E| being the number of links in E. Let |T_(k)|denote the cardinality of T_(k), then,

$\begin{matrix}{\begin{matrix}{\rho = {{E_{P{(N)}}\left\lbrack \frac{\sum\limits_{V_{i}\sim V_{j}}{\sum\limits_{r_{sd} \in R_{ij}}N_{sd}}}{E} \right\rbrack} = \frac{\sum\limits_{k = 0}^{E}{k\;{\gamma^{k}\left( {1 - \gamma} \right)}^{{E} - k}{T_{k}}}}{{E}{\sum\limits_{k = 0}^{E}{{\gamma^{k}\left( {1 - \gamma} \right)}^{{E} - k}{T_{k}}}}}}} \\{{= {\frac{\sum\limits_{k = 1}^{E}{k\;\theta^{k}{T_{k}}}}{{E}{\sum\limits_{k = 0}^{E}{\theta^{k}{T_{k}}}}}\overset{definition}{=}{\xi(\theta)}}},}\end{matrix}{{{{where}\mspace{14mu}\theta} = {\gamma/\left( {1 - \gamma} \right)}},{\theta > 0.}}} & (25) \\{\frac{\partial\left( {\xi(\theta)} \right)}{\partial\theta} = {\frac{1}{{E}\left( {\sum\limits_{k = 0}^{E}{\theta^{k}{T_{k}}}} \right)^{2}}{\left\{ {{\left( {\sum\limits_{k = 1}^{E}{k^{2}\theta^{k - 1}{T_{k}}}} \right)\left( {\sum\limits_{k = 0}^{E}{\theta^{k}{T_{k}}}} \right)} - {\left( {\sum\limits_{k = 1}^{E}{k\;\theta^{k}{T_{k}}}} \right)\left( {\sum\limits_{k = 1}^{E}{k\;\theta^{k - 1}{T_{k}}}} \right)}} \right\}.}}} & (26)\end{matrix}$Using Cauchy-Schwartz Inequality, it can be shown that∂(ξ(θ))/∂θ>0, ∀θ>0.  (27)Since ∂θ/∂γ>0, ∀0<γ<1, the following is obtained:∂ρ/∂γ>0, ∀0<γ<1.  (28)Therefore, ρ increases monotonically in γ.

Referring to Proposition 1, (a) if γ=0.5, the undirected probabilisticgraph represents a uniform probability distribution on all possible waysof using wavelength λ without violating the capacity constraint; (b) Ifγ→1, ρ increases toward the maximum value, which is determined by boththe network topology and the route set R; and (c) If γ→0, ρ approaches0. For simplicity of analysis, it is assumed that γ_(ij)≡γ,∀V_(i)˜V_(j), in the rest of the disclosure. From (22), it follows thatP(N|R _(f) =f _(sd))∝n _(sd) P(N\N _(sd) ,N _(sd)=1).  (29)

Having described the derivation of the physical and network layer modelsabove, the cross-layer model of attack propagation can be obtained bycombining the physical- and the network-layer model using a factorgraph, which corresponds to the following joint probability:P(X _(f) _(sd) ,S _(f) _(sd) ,N|R _(f) =f _(sd))=P(S _(f) _(sd) ,X _(f)_(sd) |N,R _(f) =f _(sd))P(N|R _(f) =f _(sd)),  (30)where X_(f) _(sd) =(X_(i):V_(i)∈V_(f) _(sd) ) and S_(f) _(sd)=(S_(i):V_(i)∈V_(f) _(sd) ). That is, equation (30), describes across-layer model according to one embodiment. In equation (30), P(S_(f)_(sd) ,X_(f) _(sd) |N,R_(f)=f_(sd)) corresponds to the physical layermodel, which characterizes the probability distribution of the statusand the number of affected channels at each node on the attacker's routegiven the status of each route and the source of attack. Additionally,P(N|R_(f)=f_(sd)) in equation (30) corresponds to the network layermodel, which characterizes the probability distribution of the status ofeach network route given the source of attack.

A factor graph generally refers to a bipartite graph showing how aglobal function can be factorized into a product of local functions.Each local function depends on a subset of the variables. There are twotypes of nodes in a factor graph: a variable node for each variable, anda factor node for each local function. There is an edge connecting avariable node to a factor node if and only if the variable is anargument of the local function.

FIG. 7 shows the factor graph representation 700 for the mesh network inFIG. 4 when the attack is started from flow BD. The lower portion 702 ofthe factor graph 700 represents attack propagation at the physicallayer. As the attack may propagate from node V_(i) to V_(i+1), V_(i),V_(i+1) ∈V_(f) _(sd) , X_(i) and X_(i+1) are connected to the samefactor node P(X_(i+1)|X_(i),R_(f)=f_(BD)) 710. Furthermore, the numberof affected channels at node V_(i) is determined by X_(i) and routesthat traverses node V_(i). Therefore, S_(i), X_(i), and those routespassing through node V_(i) are connected to the factor node 708 thatdescribes the conditional probability in (18).

The upper portion 704 of the factor graph 700 characterizes thedependence at the network layer. All the network routes that share acommon network link ij are connected to the clique function ψ_(ij) in(22). Here, the factor graph 700 provides an explicit representation ofthe dependencies among different network components during attackpropagation. Additional features of interest in the factor graph 700include the clique function for link AB 706.

Explaining FIG. 7 further, the graph 700 provides a cross-layer view ofdisturbance propagation in the network. The lower portion 702 of thegraph 700 corresponds to the disturbance propagation at the physicallayer corresponding to FIG. 5. As explained above, if the disturbancestarted on flow BD, it may propagate from node B to node C, and fromnode C to node D. Whether the node is affected by the attack or not isdenoted by the variables Xi (i=B, C, and D). Further, how likely theattack may propagate to the next node depends on the functional nodesbetween two adjacent nodes (e.g., component 710 in FIG. 7 determines howlikely the disturbance may propagate from node B to node C).

The upper portion 704 of the graph 700 corresponds to the network layermodel corresponding to FIG. 6, which shows how different routes in thenetwork are dependent. All the network routes that share a same networklink are connected to the same functional node. For instance, route AF,AB, AC all traverse link AB, thus the variables that denote the statusof these three routes are connected to the same function node 706.

Given the status of each network node and the status of each route thatpasses the particular node, the number of affected channels can bedecided. For example, node B is passed through by route BF, AF, AB, AC,and BC. Therefore, the variables denoting the status of these routes areconnected to the same functional node 708 with variables X_(B) andS_(B). Thus, the cross-layer model provides an efficient visualizationfor disturbance propagation in the network, characterizing thedisturbance propagation based on the network layer and the physicallayer dependencies and interactions in the optical network.

Factor graphs subsume directed and undirected probabilistic graphicalmodels, and provide explicit representations of the factorization ofprobability distributions. The application of factor graphs provides atleast two advantages: (1) it shows the intricate dependencies amongdifferent network components during a crosstalk attack; and (2) itprovides computationally efficient algorithms to evaluate the networkresilience loss, as described further below.

The cross-layer model can be used to study network resilience. Thefollowing description provides an explanation of resilience in thecontext of the physical layer and quantifies how the resilience varieswith physical topology as well as the physical layer vulnerabilities,the latter characterized by α_(i) in (16). Consider the impact ofphysical-layer vulnerabilities by considering the lower and upper boundsof network resilience loss M_(f) _(sd) . The lower bound of M_(f) _(sd)results from the best-case scenario of resilience upon attack: there areno active connections on wavelength λ that traverse link ij,∀V_(i)∈V_(f) _(sd) , V_(j)∉V_(f) _(sd) , V_(i)˜V_(j). In this case, atthe switching plane of each node along the attacker's route, only twochannels are active that correspond to the connection on the attacker'sroute. The upper bound results from the worst-case scenario of networkresilience upon attack: there always exists an active connectioninserted into the network at node V_(i) and traverses link ij,∀V_(i)∈V_(f) _(sd) , V_(j)∉V_(f) _(sd) , V_(i)˜V_(j). In this case, thenumber of active channels in the switching plane of node V_(i) is2(d_(i)−1) or 2d_(i), ∀V_(i)∈V_(f) _(sd) , where d_(i) is the nodaldegree of V_(i).

For simplicity, assume α_(i)≡α, ∀V_(i)∈V_(f) _(sd) . Then, the networkresilience loss for a given source of attack f_(sd) can be boundedaccording to Proposition 2 as follows:

$\begin{matrix}{{{k + {\sum\limits_{i = 1}^{k}\alpha^{i - 1}}} \leq M_{f_{sd}} \leq {k + {2\left( {1 + \alpha^{k - 1}} \right)} + {\sum\limits_{i = 1}^{k}{\left( {{2d_{i}} - 3} \right)\alpha^{i - 1}}}}},} & (31)\end{matrix}$where k is the total number of nodes in V_(f) _(sd) , k>1. The lowerbound in (31) characterizes the effect of route-length and α on attackpropagation, which increases polynomially with respect to α.Furthermore,

$\begin{matrix}{{k + {\sum\limits_{i = 1}^{k}\alpha^{i - 1}}} = \left\{ \begin{matrix}{{k + 1 + \alpha + {o(\alpha)}},} & {{{{as}\mspace{14mu}\alpha}->0},} \\{{{2k} - {0.5{k\left( {k - 1} \right)}\left( {1 - \alpha} \right)} + {o\left( {1 - \alpha} \right)}},} & {{{{as}\mspace{14mu}\alpha}->1},}\end{matrix} \right.} & (32)\end{matrix}$which shows that M_(f) _(sd) is determined by the length of router_(sd). The upper bound in (31) increases approximately linearly with α,where

$\begin{matrix}{\begin{matrix}{k + {2\left( {1 + \alpha^{k - 1}} \right)} +} \\{\sum\limits_{i = 1}^{k}{\left( {{2d_{i}} - 3} \right)\alpha^{i - 1}}}\end{matrix} = \left\{ \begin{matrix}{{k + {2d_{1}} - 1 + {\left( {{2d_{2}} - 3} \right)\alpha} + {o(\alpha)}},} & {{{{as}\mspace{14mu}\alpha}->{{0\mspace{14mu}{and}\mspace{14mu} k} > 2}},} \\{{k + {2d_{1}} - 1 + {\left( {{2d_{2}} - 1} \right)\alpha} + {o(\alpha)}},} & {{{{{as}\mspace{14mu}\alpha}->{0\mspace{14mu}{and}\mspace{14mu} k}} = 2},} \\{\begin{matrix}{\left( {4 - {2k}} \right) + {2{\sum\limits_{i = 1}^{k}d_{i}}} + \left\{ {{\frac{3}{2}k^{2}} - {\frac{7}{2}k} + 2 - {\sum\limits_{i = 1}^{k}{2d_{i}\left( {i - 1} \right)}}} \right\}} \\{\left( {1 - \alpha} \right) + {o\left( {1 - \alpha} \right)}}\end{matrix},} & {{{{as}\mspace{14mu}\alpha}->1},}\end{matrix} \right.} & (33)\end{matrix}$which shows that, when there is an active connection inserted into thenetwork at node V_(i) using link ij, ∀V_(i)∈V_(f) _(sd) , V_(j)∉V_(f)_(sd) , V_(i)˜V_(j), if the network vulnerability is low, M_(f) _(sd) isdetermined by the route length and the nodal degree of the source nodeof the attack. If the network vulnerability is high, M_(f) _(sd) isdetermined by the total number of network links incidental on nodesalong the attacker's flow, e.g., the number of links in set E_(f) _(sd)={e_(ij):V_(i)∈V_(f) _(sd) }. In addition,

${E_{f_{sd}}} = {{\sum\limits_{i = 1}^{k}d_{i}} + {\left( {1 - k} \right).}}$

The lower- and upper-bounds in (31) can be used to study the impact ofphysical topology on M_(f) _(sd) . For clarity, the asymptotic resultson M_(f) _(sd) in (32) and (33) for network resilience under varioustopologies are summarized below in Table I. Assume that there is onelink-shortest route between each pair of nodes in the network. Theasymptotic properties of these topologies are summarized in Table II.Combining the impacts of physical-layer vulnerability and physicaltopology, the following observations can be made:

(1) If the physical-layer vulnerability is high (α→1),

(i) The upper bound of M_(f) _(sd) shows that fully-connected meshnetworks and star networks are the least resilient due to the large sizeof the set E_(f) _(sd) .

(ii) The lower bound of M_(f) _(sd) shows that networks with a ringtopology are generally the least resilient because of the large routelength in a ring network.

(2) If the physical-layer vulnerability is low (α→0),

(i) The upper bound of M_(f) _(sd) shows that the fully-connected meshtopology is the least resilient since each node in the network has nodaldegree m−1.

(ii) The lower bound of M_(f) _(sd) shows that the ring network isgenerally the least resilient due to the large route length.

(3) Chord networks exhibit good resilience whose resilience loss M_(f)_(sd) increases logarithmically with respect to the number of nodes inthe network in the worst case.

Note that in addition to the resilience measure considered in this work,there exists other performance metrics for network resilience, e.g.two-terminal connectivity, and flexibility in route selection.Therefore, different performance metrics of network resilience are to beconsidered simultaneously when choosing a resilient network design.Overall, a chord network offers excellent resilience upon crosstalkattacks and good route selection flexibility.

TABLE I Bounds of Network Resilience Loss M_(f) _(sd) bounds Lower boundα Upper bound of M_(f) _(sd) of M_(f) _(sd) α → 1${2{\sum\limits_{i = 1}^{k}d_{i}}} + \left( {4 - {2k}} \right) - {O\left( {1 - \alpha} \right)}$2k − O(1 − α) α → 0 k + 2d₁ − 1 + O(α) k + 1 + O(α)

TABLE II Asymptotic Properties of Different Network Topologies with mNodes Ave. nodal Ave. route Ave. size of Topology degree length E_(f)_(sd) Star 1 2 m Ring 2 m/4 m/4 n-ary Tree m + 1 O(log_(n) m) O(log_(n)m) Mesh-Torus 4 O({square root over (m)}) O({square root over (m)})Fully-Connected m 1 m Mesh Chord [23] log₂ m O(log₂ m) O(log₂ m)

Having described the resilience in the context of the physical model,attention is directed now to the impact of the network layer on theresilience in terms of network load, and in particular, quantifying howthe network resilience varies jointly with the load, and thephysical-layer vulnerability α. Network resilience is determined in oneembodiment by the following equations (34) and (35). Consider the impactof network load ρ on M_(f) _(sd) . From (1),

$\begin{matrix}{{M_{f_{sd}} = {\sum\limits_{V_{i} \in V_{f_{sd}}}{E_{f_{sd}}\left\lbrack S_{i} \right\rbrack}}},} & (34)\end{matrix}$where E_(f) _(sd) [S_(i)] is the mean number of channels affected by theattack at the switching plane of node V_(i), V_(i)∈V_(f) _(sd) .Furthermore,

$\begin{matrix}{{{E_{f_{sd}}\left\lbrack S_{i} \right\rbrack} = {1 + {\alpha^{i - 1}\left\{ {1 + {\sum\limits_{{V_{i}\sim V_{j}},{V_{j} \notin V_{f_{sd}}}}\left\{ {{E_{f_{sd}}\left\lbrack {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right\rbrack} + {E_{f_{sd}}\left\lbrack {\sum\limits_{r_{ih} \in R_{ij}}N_{ih}} \right\rbrack}} \right\}}} \right\}}}},\mspace{79mu}{\forall{V_{i} \in V_{f_{sd}}}},{{where}\mspace{14mu}{E_{f_{sd}}\left\lbrack {\sum\limits_{r_{ih} \in R_{ij}}N_{ih}} \right\rbrack}}} & (35)\end{matrix}$s the mean number of active channels that are locally inserted into thenetwork at node V_(i) and leave node V_(i) through link ij, and

$E_{f_{sd}}\left\lbrack {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right\rbrack$is the mean number of active flows that enter node V_(i) through linkji, given that the attack starts from flow f_(sd). The networkresilience loss M_(f) _(sd) does not always increase with ρ forarbitrary networks with arbitrary routes. However, when practical routesets are considered, M_(f) _(sd) increases with ρ for several typicalnetwork topologies.

Explaining equations (34) and (35) further, equation (34) characterizesthe expected number of affected channels in the network given that theattack started from flow f_(sd). E_(f) _(sd) [S_(i)] is the expectednumber of channels affected by the attack at node i given that theattack is started on flow f_(sd).

$E_{f_{sd}}\left\lbrack {\sum\limits_{r_{ih} \in R_{ij}}N_{ih}} \right\rbrack$is the mean number of active channels that are locally inserted into thenetwork at node V_(i) and that leave node V_(i) through link ij, and

$E_{f_{sd}}\left\lbrack {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right\rbrack$is the mean number of active flows that enter node V_(i) through linkji, given that the attack starts from flow f_(sd). Parameter αcorresponds to the physical layer vulnerability to the disturbance, anddenotes the conditional probability that the attack may propagate to thenext downstream node on the attacker's route given that the current nodeis affected by the attack.

Referring now to the different network topologies, consider Point 1: Fora ring network, assume the route set R consists of the two-link disjointroutes between each pair of nodes in the network. Let k be the number ofnodes traversed by the attacker's flow f_(sd). Then, M_(f) _(sd)monotonically increases in ρ. In particular, M_(f) _(sd) satisfies thefollowing:

$\begin{matrix}{{{v_{1} + {2\left( {1 + \alpha^{k - 1}} \right)\gamma}} \leq M_{f_{sd}} \leq {v_{1} + {2\left( {1 + \alpha^{k - 1}} \right)\rho}}},{{{where}\mspace{14mu} v_{1}} = {k + {2{\sum\limits_{i = 1}^{k}{\alpha^{i - 1}.}}}}}} & (36)\end{matrix}$Furthermore, for 0<ρ<<1, ρ=γ+o(ρ), and the upper and the lower boundsmeetM _(f) _(sd) =v ₁+2(1+α^(k−1))ρ+o(ρ).  (37)

Detailed proof of Point 1 with regard to ring networks can be explainedas follows. Consider a ring network G(V,E) with m nodes (m>1). The routeset R consists of the two link-disjoint routes between each pair ofnodes in the network. Suppose the crosstalk attack is started on flowf_(ij) between node V_(i) and V_(j), i,j=1, 2, . . . , m, i<j. The setof nodes traversed by flow f_(ij) is V_(f) _(ij) ={V_(i), V_(i+1), . . ., V_(j)}. Then at most two nodes, node V_(i−1) and node V_(j+1), areneighbors of nodes in V_(f) _(ij) , but are not in V_(f) _(ij)themselves. Without loss of generality, one focus is on the conditionalwavelength usage at the link between V_(j) and V_(j+1).

To show that M_(f) _(ij) monotonically increases in ρ for the ringnetwork, from (35), it suffices to show that

${E_{f_{ij}}\left\lbrack {\sum\limits_{r_{uv} \in R_{{jj} + 1}}N_{uv}} \right\rbrack}\mspace{14mu}{and}\mspace{14mu}{E_{f_{ij}}\left\lbrack {\sum\limits_{r_{jh} \in R_{{jj} + 1}}N_{jh}} \right\rbrack}$monotonically increase with parameter γ in (24) for the ring network.Let

$W_{ij} = {{\sum\limits_{r_{uv} \in R_{ij}}{N_{uv}\mspace{14mu}{and}\mspace{14mu} H_{ij}}} = {\sum\limits_{r_{ih} \in R_{ij}}{N_{ih}.}}}$Then for the ring network, denote

$E_{f_{ij}}\left\lbrack {\sum\limits_{r_{uv} \in R_{{j\mspace{14mu} j} + 1}}N_{uv}} \right\rbrack$as w_(j j+1)(m,f_(ij),ring); denote

${{E_{f_{ij}}\left\lbrack {\sum\limits_{r_{jh} \in R_{{j\mspace{14mu} j} + 1}}N_{jh}} \right\rbrack}\mspace{14mu}{as}\mspace{14mu}{\varpi_{{j\mspace{14mu} j} + 1}\left( {m,f_{ij},{ring}} \right)}},$where m is the number of nodes in the ring network.

Let w₁₂(I,bus) denote the mean value of W₁₂ in an l-node network of bustopology with a route set that includes the route between each pair ofnodes, where subscript I denotes the number of nodes in the bus network.Since,w _(j j+1)(m,f _(ij),ring)=

_(j j+1)(m,f _(ij),ring)=w ₁₂(m−j+i,bus),  (38)it is sufficient to show that w₁₂(I,bus), ∀I>1, increases monotonicallywith γ.

Let θ=γ/(1−γ) and W=(W_(ij):V_(i)˜V_(j)). In addition, a configurationof (N,W) with non-zero probability is denoted as a traffic pattern. Letsum(W) denote the summation of all the components in W, then from (24),

$\begin{matrix}{{{P\left( {N,W} \right)} \propto {\theta^{{sum}{(W)}}{\prod\limits_{V_{i} \sim V_{j}}^{\;}\;{I_{2}\left( {W_{ij} = {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}} \right)}}}},} & (39)\end{matrix}$where l₂(A)=1 if A is true; and l₂(A)=0, otherwise. If (N,W) is atraffic pattern, (39) can be simplified asP(N,W)∝θ^(sum(W)).  (40)

Let T_((l),bus) denote the set of all traffic patterns on the busnetwork with I nodes (l>1). By counting all possible ways of using linkA₁A₂, the l-node bus network can be described as follows,

$\begin{matrix}{{{{P\left( {W_{12} = 1} \right)} \propto {{\theta{\sum\limits_{T_{{({I - 1})},{bus}}}\theta^{{sum}{(W_{{({I - 1})},{bus}})}}}} + {\theta^{2}{\sum\limits_{T_{{({I - 2})},{bus}}}\theta^{{sum}{(W_{{({I - 2})},{bus}})}}}} + \ldots + \theta^{I}}};}\mspace{79mu}{{P\left( {W_{12} = 0} \right)} \propto {\sum\limits_{T_{{({I - 1})},{bus}}}{\theta^{{sum}{(W_{{({k - 1})},{bus}})}}.}}}} & (41)\end{matrix}$

Let

${{f_{I}(\theta)} = {\sum\limits_{T_{{(I)},{bus}}}\theta^{{sum}{(W_{{(I)},{bus}})}}}},{\forall{I > 1}},\;{{{and}\mspace{14mu}{f_{1}(\theta)}} = 1.}$Then,P(W ₁₂=1)∝θf _(I−1)(θ)+θ² f _(I−2)(θ)+ . . . +θ^(I) ,P(W ₁₂=0)∝f_(I−1)(θ).Furthermore, the following recursive equations are observed,

$\begin{matrix}{{{{f_{1}(\theta)} = 1};{{f_{2}(\theta)} = {1 + \theta}};{{f_{i}(\theta)} = {{\left( {1 + {2\;\theta}} \right){f_{i - 1}(\theta)}} - {\theta\;{f_{i - 2}(\theta)}}}}},{i = 3},4,\ldots\mspace{11mu},{I.{Then}},} & (42) \\{{w_{12}\left( {I,{bus}} \right)} = \left\{ \begin{matrix}{\frac{\theta}{1 + \theta},} & {{{{if}\mspace{14mu} I} = 2},} \\{{1 - \frac{f_{I - 1}(\theta)}{{\left( {1 + {2\;\theta}} \right){f_{I - 1}(\theta)}} - {\theta\;{f_{I - 2}(\theta)}}}},} & {{{if}\mspace{14mu} I} > 2.}\end{matrix} \right.} & (43) \\{{Then},{\frac{\partial{w_{12}\left( {I,{bus}} \right)}}{\partial\theta} > 0},{{{for}\mspace{14mu} I} = {{2.\mspace{14mu}{If}\mspace{14mu} I} > 2}},{\frac{\partial{w_{12}\left( {I,{bus}} \right)}}{\partial\theta} = {\frac{{2{f_{I - 1}^{\prime}(\theta)}} + {\theta\left( {{f_{I - 1}^{\prime}f_{I - 2}} - {f_{I - 1}f_{I - 2}^{\prime}}} \right)}}{\left( {{\left( {1 + {2\;\theta}} \right){f_{I - 1}(\theta)}} - {\theta\;{f_{I - 2}(\theta)}}} \right)^{2}}.}}} & (44) \\{{{f_{I}^{\prime}f_{I - 1}} - {f_{I}f_{I - 1}^{\prime}}} = {{2f_{I - 1}^{2}} - {f_{I - 1}f_{I - 2}} + {{\theta\left( {{f_{I - 1}^{\prime}f_{I - 2}} - {f_{I - 1}f_{I - 2}^{\prime}}} \right)}.}}} & (45)\end{matrix}$Since f₂′f₁−f₂f₁′=2θ²+4θ+1>0, through mathematical induction, from (45),the following is observed f_(I)′f_(I−1)−f_(I)f_(I−1)′>0, ∀I>1, and

$\begin{matrix}{{\frac{\partial{w_{12}\left( {I,{bus}} \right)}}{\partial\theta} > 0},{\forall{I > 1.}}} & (46)\end{matrix}$Since θ=γ/(1−γ), and 0<θ<1, it follows that

${\frac{\partial{w_{12}\left( {I,{bus}} \right)}}{\partial\gamma} > 0},{\forall{I > 1.}}$From Proposition 1, M_(f) _(ij) monotonically increases in ρ for thering network.

The upper and lower bound of M_(f) _(sd) in (36) is obtained by showingthatγ≦w _(j j+1)(m,f _(ij),ring)≦ρ.  (47)

For an arbitrary network topology G(V,E), if r_(ij)∈R, i.e., there isone route from node V_(i) to V_(j) in R, where V_(i)∈V_(f) _(sd) ,V_(j)∉V_(f) _(sd) , and V_(i)˜V_(j), then,E_(f) _(sd) [W_(ij)]≧γ,  (48)Hence, the ring network considered here satisfies the condition in (48).

Let E₁ be the set of links that are not traversed by flow f_(sd):E₁={e_(ij):V_(i)˜V_(j), r_(sd)∉R_(ij)}. Let R₁ be the set of routes in Rthat only traverse links in E₁. Let E₂=E₁\{e_(ij)}, and R₂ be the set ofroutes in R that only traverse links in E₂. Hence, R₂⊂R₁⊂R, if r_(ij)∈R.

Let T_(E) ₁ ={(N_(E) ₁ ,W_(E) ₁ )} be the set of traffic patternsrestricted to a network formed by link set E₁ with route set R₁. LetT_(E) ₂ ={(N_(E) ₂ ,W_(E) ₂ )} be the set of traffic patterns restrictedto a network formed by link set E₂ with route set R₂. Then,

$\begin{matrix}{{{{E_{f_{sd}}\left\lbrack W_{ij} \right\rbrack} = \frac{{\theta\;{Z_{1}(\theta)}} + {Z_{2}(\theta)}}{{\left( {1 + \theta} \right){Z_{1}(\theta)}} + {Z_{2}(\theta)}}},{where}}{{{Z_{1}(\theta)} = {\sum\limits_{T_{E_{2}}}\theta^{{sum}{(W_{E_{2}})}}}},{{Z_{2}(\theta)} = {{\sum\limits_{T_{E_{1}}}\theta^{{sum}{(W_{E_{1}})}}} - {\left( {1 + \theta} \right){{Z_{1}(\theta)}.}}}}}} & (49)\end{matrix}$In addition, Z₂(θ)>0 if there is a route that traverses link ij and oneor more links in set E₂; Z₂(θ)=0, otherwise.

Since Z₁(θ)>0, the following is observed

$\begin{matrix}{{{E_{f_{sd}}\left\lbrack W_{ij} \right\rbrack} \geq \frac{\theta}{1 + \theta}} = {\gamma.}} & (50)\end{matrix}$To show that w_(j j+1)(m,f_(ij),ring)≦ρ, from (38), it suffices to showthatw ₁₂(m−j+1,bus)≦ρ,  (51)which can be proved through the following two lemmas.w ₁₂(I,bus)≦w ₁₂(I+1,bus), ∀I>1.  Lemma 1:w ₁₂(m,bus)≦ρ, ∀m>1.  Lemma 2:Lemma 1 and 2 are proved using induction similarly as in the proof of(46). Detailed proof is omitted here. Using (47), (36) can be obtainedfrom (35).

Point 2: For a star network, assume that the route set R consists of theroutes between each pair of nodes in the network. Let m, m>1, be thenumber of nodes in the network. Let the hub node be denoted as V_(m).Then, M_(f) _(sd) monotonically increases in ρ. In particular, M_(f)_(sd) satisfies

$\begin{matrix}{M_{f_{sd}} \geq \left\{ \begin{matrix}{{3 + \alpha + {\left( {m - 2} \right)\alpha\;\gamma}},} & {{{{if}\mspace{14mu} f_{sd}} = f_{A_{i}A_{m}}},{i = 1},\ldots\mspace{11mu},{m - 1},} \\{{3 + \alpha + {\left( {m - 2} \right)\gamma}},} & {{{{if}\mspace{14mu} f_{sd}} = f_{A_{m}A_{i}}},{i = 1},\ldots\mspace{11mu},{m - 1},} \\{{4 + \alpha + \alpha^{2} + {\left( {m - 3} \right)\alpha\;\gamma}},} & {{otherwise},}\end{matrix} \right.} & (52) \\{M_{f_{sd}} \leq \left\{ \begin{matrix}{{3 + \alpha + {2\left( {m - 2} \right)\alpha\;\rho}},} & {{{{if}\mspace{14mu} f_{sd}} = f_{A_{i}A_{m}}},{i = 1},\ldots\mspace{11mu},{m - 1},} \\{{3 + \alpha + {2\left( {m - 2} \right)\rho}},} & {{{{if}\mspace{14mu} f_{sd}} = f_{A_{m}A_{i}}},{i = 1},\ldots\mspace{11mu},{m - 1},} \\{{4 + \alpha + \alpha^{2} + {2\left( {m - 3} \right)\alpha\;\rho}},} & {{otherwise}.}\end{matrix} \right.} & (53)\end{matrix}$

Furthermore, for 0<ρ<<1, the bounds are tight, and

$\begin{matrix}{M_{f_{sd}} = \left\{ \begin{matrix}{{3 + \alpha + {2\left( {m - 2} \right)\alpha\;\rho} + {o(\rho)}},} & {{{{if}\mspace{14mu} f_{sd}} = f_{A_{i}A_{m}}},{i = 1},\ldots\mspace{11mu},{m - 1},} \\{{3 + \alpha + {2\left( {m - 2} \right)\rho} + {o(\rho)}},} & {{{{if}\mspace{14mu} f_{sd}} = f_{A_{m}A_{i}}},{i = 1},\ldots\mspace{11mu},{m - 1},} \\{{4 + \alpha + \alpha^{2} + {2\left( {m - 3} \right)\alpha\;\rho} + {o(\rho)}},} & {{otherwise},}\end{matrix} \right.} & (54)\end{matrix}$

Proofs of Point 2 with regard to star networks can be explained asfollows. For a network of star topology with m nodes, m>2, and a routeset R that consists of the routes between each pair of nodes. Let nodeV_(m) be the hub node of the star network. Let

$W_{ij} = {{\sum\limits_{r_{uv} \in R_{ij}}{N_{uv}\mspace{14mu}{and}\mspace{14mu} H_{ij}}} = {\sum\limits_{r_{ih} \in R_{ij}}{N_{ih}.}}}$When the attack is started on flow f_(1m), w_(mi)(m,f_(1m),star)+

_(mi)(m,f_(1m),star), i=2, . . . , m−1, increases monotonically with γ,where

${{w_{m\; i}\left( {m,f_{1m},{star}} \right)} = {E_{f_{1m}}\left\lbrack {\sum\limits_{r_{uv} \in R_{m\; i}}N_{uv}} \right\rbrack}},{and}$${\varpi_{m\; i}\left( {m,f_{1m},{star}} \right)} = {{E_{f_{1m}}\left\lbrack {\sum\limits_{r_{mh} \in R_{m\; i}}N_{mh}} \right\rbrack}.}$

Let T_((I),star) denote the set of all traffic patterns on the starnetwork with I nodes. By counting all possible ways of using link mi, itcan be found that, for the l-node star network,

$\begin{matrix}{{{P\left( {W_{mi} = {{1\text{❘}R_{f}} = f_{1m}}} \right)} \propto {{\theta{\sum\limits_{T_{{({m - 1})},{star}}}\theta^{{sum}{(W_{{({m - 1})},{star}})}}}} + {\left( {m - 2} \right)\theta^{2}{\sum\limits_{T_{{({m - 2})},{star}}}\theta^{{sum}{(W_{{({m - 2})},{star}})}}}}}};} & (55) \\{{{P\left( {W_{mi} = {{0\text{❘}R_{f}} = f_{1m}}} \right)} \propto {\sum\limits_{{T{({m - 1})}},{star}}\theta^{{sum}{(W_{{({m - 1})},{star}})}}}};} & (56) \\{{{P\left( {H_{mi} = {{1\text{❘}R_{f}} = f_{1m}}} \right)} \propto {\theta{\sum\limits_{T_{{({m - 1})},{star}}}\theta^{{sum}{(W_{{({m - 1})},{star}})}}}}};} & (57) \\{{P\left( {H_{mi} = {{0\text{❘}R_{f}} = f_{1m}}} \right)} \propto {{\sum\limits_{T_{{({m - 1})},{star}}}\theta^{{sum}{(W_{{({m - 1})},{star}})}}} + {\left( {m - 2} \right)\theta^{2}{\sum\limits_{T_{{({m - 2})},{star}}}{\theta^{{sum}{(W_{{({m - 2})},{star}})}}.}}}}} & (58)\end{matrix}$Let

${t_{1} = {{1\mspace{14mu}{and}\mspace{14mu}{t_{l}(\theta)}} = {\sum\limits_{T_{{(l)},{star}}}\theta^{{sum}{(W_{{(l)},{star}})}}}}},{l > 1.}$Then, the following recursive equations are observed,t ₂=1+θ; t ₁=(1+θ)t _(I−1)+(I−2)θ² t _(k−2) , ∀I>2.Therefore, from (55)-(58),

$\begin{matrix}{{{w_{mi}\left( {m,f_{1m},{star}} \right)} + {\varpi_{mi}\left( {m,f_{1m},{star}} \right)}} = {1 + {\frac{\left( {\theta - 1} \right)t_{m - 1}}{{\left( {1 + \theta} \right)t_{m - 1}} + {\left( {m - 2} \right)\theta^{2}t_{m - 2}}}.}}} & (59)\end{matrix}$

Through induction similarly as in the proof of (46),

$\frac{\partial\left\{ {{w_{mi}\left( {m,f_{1m},{star}} \right)} + {\varpi_{mi}\left( {m,f_{1m},{star}} \right)}} \right\}}{\partial\gamma} > 0.$Similarly, when the attack is started from flow f_(A) ₁ _(A) ₂ , it canbe shown that

$\frac{\partial\left\{ {{w_{mi}\left( {m,f_{12},{star}} \right)} + {\varpi_{mi}\left( {m,f_{12},{star}} \right)}} \right\}}{\partial\gamma} > 0.$Thus, from (35), it follows that M_(f) _(if) monotonically increases inρ for the star network. The upper and lower bound of M_(f) _(sd) in (52)and (53) is obtained by showing thatγ<w _(mi)(m,f _(1m),star)+

_(mi)(m,f _(1m),star)≦2ρ,γ<w _(mi)(m,f ₁₂,star)+

_(mi)(m,f ₁₂,star)≦2ρ.  (60)Since

_(mi)(m,f _(1m),star)≦w _(mi)(m,f _(1m),star)and

_(mi)(m,f ₁₂,star)≦w _(mi)(m,f ₁₂,star),  (61)Equation (60) can be obtained by showingγ≦w _(mi)(m,f _(1m),star)≦ρ,γ≦w _(mi)(m,f ₁₂,star)≦ρ.  (62)The proof of (62) is similar to that of (47), and is omitted.

Referring to point 2, M_(f) _(sd) generally is the sum of two terms:e.g. in (62) below, (3+α) that corresponds to the number of affectedchannels used by flows on the attacker's route; and (2(m−2)αρ+o(ρ)) thatcorresponds to the number of affected channels used by flows not on theattacker's route.

M_(f) _(sd) is compared for ring and star networks. In both cases, M_(f)_(sd) is linearly increasing in ρ for 0<ρ<<1. However, for ringnetworks, M_(f) _(sd) is polynomially increasing in α; whereas for starnetworks, M_(f) _(sd) is linearly increasing in α. For ring networks,M_(f) _(sd) is linearly increasing in k (the number of nodes in V_(f)_(sd) ). For star networks, M_(f) _(sd) is linearly increasing in m (thenumber of nodes in the network).

In the description that follows, a focus is placed on the impact ofnetwork load ρ on the average network resilience loss (M), which is themean value of network resilience loss over all possible source ofattacks. Consider a ring network with m, m>1, nodes, V₁,V₂, . . . ,V_(m), and a route set R that includes all the two link-disjoint pathsbetween each pair of nodes in the network. Then, the following equationscan be presented:

Point 3:

$\begin{matrix}{{M_{{ring},m} = {\frac{1}{m - 1}{\sum\limits_{i = 1}^{m - 1}{a_{i}M_{f_{{1i} + 1}}}}}},} & (63)\end{matrix}$where a_(i)=P(N_(1i+1)=1) is the probability that a connection with ilinks between two terminal nodes, V₁ and V_(i+1), is active, and M_(f)_(1i+1) is the network resilience loss when the attack is started fromflow f_(1i+1).

Furthermore, the following can be presented:

$\begin{matrix}{\mspace{79mu}{{a_{i} = {\theta^{i}{f_{m - i + 1}/g_{m}}}},}} & (64) \\{\mspace{79mu}{{\theta = {\gamma/\left( {1 - \gamma} \right)}},}} & (65) \\{{f_{m} = {{\frac{\sqrt{1 + {4\theta^{2}}} + 1}{2\sqrt{1 + {4\theta^{2}}}}\left( \frac{1 + {2\theta} + \sqrt{1 + {4\theta^{2}}}}{2} \right)^{m - 1}} + {\frac{\sqrt{1 + {4\theta^{2}}} - 1}{2\sqrt{1 + {4\theta^{2}}}}\left( \frac{1 + {2\theta} - \sqrt{1 + {4\theta^{2}}}}{2} \right)^{m - 1}}}},} & (66) \\{\mspace{79mu}{{g_{m} = {f_{m} + {\sum\limits_{j = 1}^{m - 1}{j\;\theta^{j}f_{m + 1 - j}}}}},{m > 1.}}} & (67)\end{matrix}$Detailed proof of Point 3 can be explained as follows. First,a_(i)=P(N_(1i+1)=1) is described. Through solving the differenceequation in (42), (66) can be proved.

Let T_((I),ring) be the set of all traffic patterns on a ring networkwith I nodes and a route set that includes all possible link-disjointshortest paths between each pair of nodes in the network. Let

$g_{m} = {\sum\limits_{T_{{(m)},{ring}}}{\theta^{{sum}{(W_{{(m)},{ring}})}}.}}$By counting different ways of using one single link in the ring network,the following is observed:g _(k) =f _(k) +θf _(k)+2θf _(k−1)+ . . . +(k−1)θ^(k−1) f ₂.  (68)Therefore, a_(i)=P(N_(A) ₁ _(A) _(i+1) =1)=θ^(i)f_(k−i+1)/g_(k). Usingthe lower and upper bound of M_(f) _(sd) for the ring network in (36), 3is obtained.

Using Point 1, the following bounds are observed:

$\begin{matrix}{{M_{{ring},m} \geq {\frac{1}{\left( {m - 1} \right)}{\sum\limits_{i = 1}^{m - 1}\left\{ {a_{i}\left( {i + 1 + {\sum\limits_{j = 0}^{i}\alpha^{j}} + {2\left( {1 + \alpha^{i}} \right)\gamma}} \right)} \right\}}}},} & (69) \\{{M_{{ring},m} \leq {\frac{1}{\left( {m - 1} \right)}{\sum\limits_{i = 1}^{m - 1}\left\{ {a_{i}\left( {i + 1 + {\sum\limits_{j = 0}^{i}\alpha^{j}} + {2\left( {1 + \alpha^{i}} \right)\rho}} \right)} \right\}}}},} & (70)\end{matrix}$

The difference between the upper and the lower bound of M_(ring,m) isO((ρ−γ)/m). Furthermore, (63) can be simplified as follows:

$\begin{matrix}{{M_{{ring},m} = {{\rho\;{M_{f_{A_{1}A_{2}}}/\left( {m - 1} \right)}} + {o(\rho)}}},{{{as}\mspace{14mu}\rho}->0.}} & (71) \\{{M_{{ring},m} = {\sum\limits_{i = 1}^{m - 1}{\frac{1}{2^{i}}{M_{f_{A_{1}A_{i + 1}}}/\left( {m - 1} \right)}}}},{{{as}\mspace{14mu}\rho}->1},{m->{\infty.}}} & (72)\end{matrix}$

Then, the following can be described:M _(ring,m)=ρ(3+α)/(m−1)+o(ρ), as ρ→0,  (73)which shows that, when the network load is low, M_(ring,m) increasesalmost linearly with ρ and α; and is in the order of O(ρ/m).

When the network load is high, the following can be described:

$\begin{matrix}{M_{{ring},m} = {\frac{1}{m - 1}\left\{ {{\sum\limits_{i = 1}^{m - 1}{\frac{1}{2^{i}}\left( {i + 1 + {\sum\limits_{j = 0}^{i}\alpha^{j}} + {2\left( {1 + \alpha^{i}} \right)}} \right\}}},{{{as}\mspace{14mu}\rho}->1},{m->{\infty.}}} \right.}} & (74)\end{matrix}$

Furthermore, if α=1, (74) can be simplified as follows:

$\begin{matrix}{{M_{{ring},m} = {\frac{1}{m - 1}\left( {10 - \frac{{2m} + 8}{2^{m - 1}}} \right)}},} & (75)\end{matrix}$which shows that M_(ring,m) is in the order of O(1/m).

Next consider a star network with m nodes, V₁,V₂, . . . , V_(m), whereV_(m) is the hub node of the star network; and a route set R thatincludes all the link-disjoint paths between each pair of nodes in thenetwork. It follows that,

Point 4:

$\begin{matrix}{{M_{{star},m} = \frac{{b_{1}\left( {M_{f_{A_{1}A_{k}}} + M_{f_{A_{k}A_{1}}}} \right)} + {2{b_{2}\left( {m - 2} \right)}M_{f_{A_{1}A_{2}}}}}{2\left( {m - 1} \right)}},{m > 3},} & (76)\end{matrix}$where b₁=P(N_(A) ₁ _(A) _(k) =1), b₂=P(N_(A) ₁ _(A) ₂ =1) withb ₁ =θt _(m−1) /t _(m) ; b ₂=θ² t _(m−2) /t _(m) ; t ₁=1; t ₂=1+θ; t_(i)=(1+θ)t _(i−1)+(i−2)θ² t _(i−2) , ∀i>2.

Detailed proof of Point 4 can be explained as follows. From (55)-(58),P(N _(1m)=1)=θt _(k−1) /t _(k) ; P(N _(A) ₁ _(A) ₂ =1)=θ² t _(m−2) /t_(m) ; ∀k>3.Using the lower and upper bound of M_(f) _(sd) for the star network in(52) and (53), Point 4 can be obtained.

Furthermore, using Point 2, the following bounds for M_(star,m) areobserved

$\begin{matrix}{{M_{{star},m} \geq {\frac{1}{2\left( {m - 1} \right)}\left\{ {{b_{1}\left( {4 + {\left( {1 + \alpha} \right)\left( {2 + {\left( {m - 2} \right)\gamma}} \right)}} \right)} + {2{b_{2}\left( {m - 2} \right)}\left( {4 + \alpha + \alpha^{2} + {\left( {m - 3} \right){\alpha\gamma}}} \right)}} \right\}}},} & (77) \\{M_{{star},m} \leq {\frac{1}{2\left( {m - 1} \right)}{\left\{ {{b_{1}\left( {4 + {2\left( {1 + \alpha} \right)\left( {1 + {\left( {m - 2} \right)\rho}} \right)}} \right)} + {2{b_{2}\left( {m - 2} \right)}\left( {4 + \alpha + \alpha^{2} + {2\left( {m - 3} \right){\alpha\rho}}} \right)}} \right\}.}}} & (78)\end{matrix}$

The difference between the upper and the lower bound of M_(star,m) isO(α(2ρ−γ)) when m is large, since b₂ is O(1/m). In addition, when thenetwork load is low,M _(star,m)=ρ(3+α)/m+o(ρ), as ρ→0,  (79)which shows that M_(star,m) is O(ρ/m); and increases linearly with α.

When the network load is high (ρ−1), the following can be presented:M _(star,m) =O(αρ), as ρ−1,  (80)which shows that, when the star network is under high load, M_(star,m)increases linearly with α.

For a general network G(V,E) with a fixed set of route R, the followingupper bound for M can be described:

Point 5:

$\begin{matrix}{{M \leq {\frac{1}{R}{\max\limits_{f_{sd}}{\left\{ M_{f_{sd}} \right\}\rho{E}}}}},} & (81)\end{matrix}$where |E| is the cardinality of the set of edges in the network; |R| isthe cardinality of the route set R. The proof of point 5 can beexplained as follows. Since

${M = {\sum\limits_{f_{sd}}\;{M_{f_{sd}}{{P\left( {N_{sd} = 1} \right)}/2}{R}}}},$the following is observed:

$\begin{matrix}{M \leq {\frac{\;{\sum\limits_{f_{sd}}\;{P\left( {N_{sd} = 1} \right)}}}{2{R}}{\max\limits_{f_{sd}}{\left\{ M_{f_{sd}} \right\}.}}}} & (82)\end{matrix}$

Let N=(N_(sd):r_(sd)∈R), and E [ ] stands for expectation. Since eachconnection consists of two flows, then

${\sum\limits_{f_{sd}}\;{P\left( {N_{sd} = 1} \right)}} = {2\;{{{sum}(N)}.}}$Thus, to prove Point 5, it suffices to show thatE(sum(N))≦ρ|E|.The following is observed:

${{E\left\lbrack {{sum}\;(N)} \right\rbrack} = {\sum\limits_{w}\;{{E\left\lbrack {{sum}\;(N)\text{|}W} \right\rbrack}{P(W)}}}},{where}$W = (W_(ij), i ∼ j).Since E [sum(N)|W]≦E [sum(W)|W],E(sum(N))≦E(sum(W)).  (83)As E [sum(W)]=ρ|E|, from (83),E(sum(N))≦ρ|E|.It follows that

$M \leq {\frac{1}{R}{\max\limits_{f_{sd}}{\left\{ M_{f_{sd}} \right\}\rho{{E}.}}}}$

In (81), ρ|E|/|R| corresponds to the upper bound of the probability thatthe crosstalk attack occurs in the network, and is accurate when thenetwork route set R only consists of routes with link-length 1. Thebound in (81) provides a worst case estimation of M.

Furthermore, suppose that all the routes in the set R are of the samelink length l. Then the probability that a crosstalk attack happens inthe network is (ρ|E|)/(l|R|), and (81) can be refined as follows:

$\begin{matrix}{M \leq {\frac{1}{{R}l}{\max\limits_{f_{sd}}{\left\{ M_{f_{sd}} \right\}\rho{{E}.}}}}} & (84)\end{matrix}$

When the length of each network route and the network resilience lossM_(f) _(sd) are the same for each possible source of attack, theequality in (84) holds. Point 5 suggests the upper bound of averagenetwork resilience loss is affected by the following factors:

(1) The network load in the network. The upper bound in (84) increasesat least linearly with ρ.

(2) The number of links in the network. The larger the number of linksin the network, the less resilient the network. The upper bound in (84)increases linearly with the number of links in the network.

(3) The number of routes in the network. The larger the number of routesin the network, the more resilient the network. This is because theprobability for a route to be chosen as the attacker's route is smaller.

Next Equation (81) is used to study a mesh-torus network with m nodesand a route set R, which includes: (1) the unique link-shortest routebetween each pair of nodes if applicable; and (2) one shortest routebetween each pair of nodes, which forms the border of the sub-grid withthe two nodes at the diagonally opposite corners.

Point 6 can be expressed as follows:

$\begin{matrix}{{\max\limits_{f_{sd}}\left\{ M_{f_{sd}} \right\}} \leq \left\{ \begin{matrix}{{\frac{6\left( {1 - \alpha^{\sqrt{m} + 1}} \right)}{\left( {1 - \alpha} \right)} + 4},} & {{{{if}\mspace{14mu}\alpha} \neq 1},} \\{{{6\sqrt{m}} + 4},} & {{otherwise}.}\end{matrix} \right.} & (85)\end{matrix}$

Then, from (81),

$\begin{matrix}{{M_{{torus},m} \leq {\frac{2m\;\rho}{m\left( {m - 1} \right)}{\max\limits_{f_{sd}}\left\{ M_{f_{sd}} \right\}}}},{M_{{torus},m} \leq \left\{ \begin{matrix}{{\frac{2\rho}{\left( {m - 1} \right)}\left( {\frac{6\left( {1 - \alpha^{\sqrt{m} + 1}} \right)}{\left( {1 - \alpha} \right)} + 4} \right)},} & {{{{if}\mspace{14mu}\alpha} \neq 1},} \\{{{\frac{2\rho}{\left( {m - 1} \right)}6\sqrt{m}} + 4},} & {{otherwise}.}\end{matrix} \right.}} & (86)\end{matrix}$

Furthermore, when ρ→0, it can be found thatM _(torus,m)=ρ(3+α)/(m−1)+o(ρ), as ρ→0.  (87)

A comparison of the average network resilience loss for ring, star andmesh networks is shown in Table III below. The following can beobserved:

(1) When the network load is low (0<ρ<<1), M is O(ρ/m). This is becausewhen the load is close to zero (0), the network is most likely in eitherof two states: (a) there is no active connection in the network; or (b)there is an active connection of link length 1. Specifically, withprobability O(ρ), the attack is started on a route of link length 1;with probability o(ρ), the attack is started on a route of longerlengths. For instance, as each route is the attacker's route with equalprobability, the attack starts on routes of link length 1 in themesh-torus network with probability 2ρ/(m−1) and M_(f) _(sd) =(3+α)+o(ρ)if ρ<<1.

(2) When the network load is high (ρ→1), the star network is the leastresilient, with M being O(α). This is because, for the star network,nodes in the set V_(f) _(sd) , ∀r_(sd)∈R, has the most number ofneighboring links. Ring and mesh-torus networks show good networkresilience in O(ρ/m).

TABLE III Average Network Resilience Loss (M) M ρ → 0 ρ → 1 Ring networkρ(3 + α)/(m − 1) O(1/m) Star network ρ(3 + α)/m O(α) Mesh-torus 2ρ(3 +α)/(m − 1) $\left\{ \begin{matrix}{{O\left( {{1/\left( {1 - \alpha} \right)}m} \right)},} & {{{{if}\mspace{14mu}\alpha} \neq 1},} \\{{O\left( {1/\sqrt{m}} \right)},} & {{otherwise}.}\end{matrix}\quad \right.$

For networks with irregular topologies, the sum-product algorithm on thefactor graph is used. The sum-product algorithm is then compared withthe exact resilience calculation through enumerations of all networktraffic patterns. Enumeration has the computational complexityexponential in the number of routes, and is thus not applicable tonetworks with even a medium number of routes. The computationalcomplexity of the sum-product algorithm is exponential in the maximumnodal degree of the factor graph for the worst case, and is thus muchmore efficient than enumeration. The sum-product algorithm providesexact results when the factor graph has no loops, and providesapproximate results otherwise.

When there are a large number routes in the set R, to further reduce thecomputational complexity of the sum-product algorithm, the followingintermediate variables can be introduced:

$\begin{matrix}{{W_{ij} = {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}}},{W_{ij} \in \left\{ {0,1} \right\}},} & (1)\end{matrix}$which is the number of flows that enter the switching plane of nodeV_(i) through link ij;

$\begin{matrix}{{H_{ij} = {\sum\limits_{r_{ih} \in R_{ij}}\; N_{ih}}},{H_{ij} \in \left\{ {0,1} \right\}},} & (2)\end{matrix}$which is the number of flows locally originated at node i and leave nodeV_(i) through link ij. Then the factor graph representation can betransformed accordingly. On the other hand, it is possible to transformfactor graphs with loops into loop-free factor graphs, so that exactresults can be obtained using the sum-product algorithm, sometimes atthe cost of computational complexity.

Consider the three networks shown in FIG. 8A-8C. Shown is a ring network802, double-ring network 804, and a mesh network 806. In each network,consider that the route set has 21 routes, which corresponds to onelink-shortest route between each pair of nodes. Using the sum-productalgorithm, network resilience loss is computed given the source ofattack M_(f) _(sd) for each f_(sd). Then, the sum-product algorithm isused to find the probability of P(N_(sd)=1). Finally, (2) is used tocompute the average network resilience loss.

FIG. 9 provides a graph 900 that depicts the relationship between ρ andaverage network resilience loss M for the networks 802 (ring), 804(double ring), and 806 (mesh) in 8A-8C, respectively, with α=0.6. It canbe observed that:

(1) M monotonically increases with ρ, in networks with all-to-alltraffic and link-shortest path routing. Moreover, for low loads, Mincreases linearly with ρ.

(2) The sum-product algorithm results in an almost exact M for the meshand ring networks, even though the factor graph representations containloops. The performance of sum-product algorithm is not as accurate, yetacceptable for the double-ring network. This suggests that thesum-product algorithm can be used for large networks where exactcalculation of resilience is infeasible.

The sum-product algorithm can be used to study the network resiliencefor the well-known benchmark National Science Foundation Network (NSF)network topology with 14 nodes and 21 bi-directional links. Assume thatthere is one link-shortest route between each pair of nodes in thenetworks. Then, there are 91 routes in R. The corresponding factor graphrepresentation contains loops, and thus the sum-product algorithmprovides an approximation for M. FIG. 10 shows a graph 1000 thatreflects the relationship between ρ and M for the NSF network topologywith α=0.3, 0.6, 0.9. The graph 1000 in FIG. 10 suggests that, if theset of network routes consists of one link-shortest route between eachpair of nodes in the network, M generally increases with the networkload. Furthermore, when the network load is low, M increases linearlywith ρ.

As described above, several factors from both the physical- and thenetwork layer that affect the resilience have been explored. Factorsfrom the physical-layer include: (1) the physical-layer vulnerability,parameters in Bayesian Belief Network that characterize how likely theattack propagates, and (2) the physical topology. Factors from thenetwork layer include active network connections that are characterizedusing network load, e.g., the probability that the wavelength, on whichthe attack is initiated, is used in the network. For all the topologiesdisclosed herein, it has been demonstrated that the average networkresilience loss increases linearly with respect to the physical-layervulnerability and light network load under link-shortest routing, andall-to-all traffic. In addition, ring and mesh-torus network show goodresilience, which are inversely proportional to the number of the nodesin the network. Numerical results also suggest that for networks withlink-shortest routing and all-to-all traffic, the network resilienceloss increases at least linearly with respect to the network load.

There are several benefits resulting from the cross-layer model based ongraphical models. For instance, the cross-layer model provides anexplicit representation of the dependencies and interactions between thephysical- and the network layer. In addition, the cross-layer modelfacilitates the analytical investigation of network resilience for ring,star, and special cases of mesh topologies. Further, the cross-layermodel facilitates the implementation of computationally efficientapproaches, e.g. the sum-product algorithm, for evaluating networkresilience. Compared to previous work by the inventors reflected by theMarch 2005 publication, “Probabilistic Graphical Models for Resilienceof All-Optical Networks under Crosstalk Attacks,” the determination ofresilience as described herein is more meaningful. For instance, in thevarious embodiments described herein, resilience is measured based onthe number of affected wavelength channels at each node. Hence, themeasurements reveal not only whether a network node is affected by anattack or not, but also quantifies how many wavelength channels areaffected at a particular node. In other words, the measure of resilienceas herein described characterizes the level of damage to each affectednode. This quantifying measure is revealed in FIG. 7, where the variableS at the middle of the factor graph 700 combines both the status of eachroute (network layer model) and the status of each node (physical layermodel). In contrast, the March 2005 publication discloses a method thatmeasures the resilience based on the percentage of nodes affected in thenetwork, which does not provide such quantifying measures orcharacterization of node damage.

In view of the above description, it should be appreciated that onemethod embodiment 200 a, as shown in FIG. 11, comprises for a physicalmodel, modeling propagation of a disturbance in an optical network understatic network conditions based on the disturbance propagation having athreshold effect only on nodes along a route followed by the disturbance(1102), for a network model, modeling a status of each network route inthe optical network based on the disturbance (1104), and combining thephysical layer model and the network layer model to provide a crosslayer model that characterizes the disturbance propagation based onnetwork layer and physical layer dependencies and interactions in theoptical network (1106).

Any process descriptions or blocks in flow charts should be understoodas representing modules, segments, or portions of code which include oneor more executable instructions for implementing specific logicalfunctions or steps in the process, and alternate implementations areincluded within the scope of the preferred embodiments of the presentdisclosure in which functions may be executed out of order from thatshown or discussed, including substantially concurrently or in reverseorder, depending on the functionality involved, as would be understoodby those reasonably skilled in the art of the present disclosure.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations,merely set forth for a clear understanding of the principles of thedisclosure. Many variations and modifications may be made to theabove-described embodiment(s) of the disclosure without departingsubstantially from the spirit and principles of the disclosure. All suchmodifications and variations are intended to be included herein withinthe scope of this disclosure and the present disclosure and protected bythe following claims.

1. A computer-implemented method for optical network evaluation,comprising: for a physical model, modeling propagation of a disturbancein an optical network under static network conditions based on thedisturbance propagation having a threshold effect only on nodes along aroute followed by the disturbance, the modeling implemented by acomputer; for a network model, modeling a status of each network routein the optical network based on the disturbance, the modelingimplemented by the computer; and combining the physical layer model andthe network layer model to provide a cross layer model thatcharacterizes the disturbance propagation based on network layer andphysical layer dependencies and interactions in the optical network, thecombining implemented by the computer, wherein modeling the statuscomprises determining a probability distribution of active connectionsin the optical network and the number of affected wavelength channels,the modeling implemented by the computer; wherein all active wavelengthchannels in the switching plane of node V_(i) are affected if node V_(i)is affected by the disturbance, wherein$\sum\limits_{V_{i} - V_{i}}\left\{ {{\sum\limits_{r_{uv} \in R_{ij}}n_{uv}} + {\sum\limits_{r_{ih} \in R_{ij}}n_{ih}}} \right\}$corresponds to the total number of active wavelength channels in theswitching plane of node V_(i), $\sum\limits_{r_{uv} \in R_{ij}}n_{uv}$corresponds to the number of flows that enter the switching plane ofnode V_(i) through link ij, and $\sum\limits_{r_{ih} \in R_{ij}}n_{ih}$corresponds to the number of flows that are locally originated from nodeV_(i) and enter the network through link ij.
 2. The method of claim 1,further comprising determining resilience of the optical network basedon the cross layer model, the determining implemented by the computer.3. The method of claim 1, wherein the physical model is represented by adirected graph.
 4. The method of claim 1, wherein the network model isrepresented by an undirected probabilistic graph.
 5. The method of claim1, further comprising displaying the cross layer model as a factor graphrepresentation.
 6. The method of claim 1, wherein the disturbancecomprises crosstalk.
 7. The method of claim 1, wherein the opticalnetwork is an all optical network.
 8. The method of claim 1, wherein thephysical layer model comprises the following equation:${{P\left( {X_{f_{sd}},{{S_{f_{sd}}\text{|}N} = n},{R_{f} = f_{sd}}} \right)} = {\prod\limits_{i = 1}^{k - 1}\;{{P\left( {{X_{i + 1}\text{|}X_{i}},{R_{f} = f_{sd}}} \right)}{\prod\limits_{i = 1}^{k}\;{P\left( {{S_{i}\text{|}X_{i}},{N = n},{R_{f} = f_{sd}}} \right)}}}}},$where S_(f) _(sd) =(S_(i):V_(j)∈V_(f) _(sd) ), and X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ) k is the number of nodes in V_(f) _(sd) ,wherein the physical layer model includes the probability of the numberof channels affected S at each network node V given the status X ofnetwork routes n and the source of attack R_(f), where R_(f) denotes theunidirectional flow f_(sd) where the attack originates.
 9. The method ofclaim 8, wherein the network layer model comprises the followingequation:${{P(N)} = {\frac{1}{Z_{N}}{\prod\limits_{({V_{i}\sim V_{j}})}\;{\gamma_{ij}{\sum\limits_{r_{uv} \in R_{ij}}\;{{N_{uv}\left( {1 - \gamma_{ij}} \right)}^{({1 - {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}}})}{l_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}} \right)}}}}}}},\mspace{79mu}{{{{where}\mspace{14mu}{l_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)}} = {{1\mspace{14mu}{if}\mspace{14mu}{\sum\limits_{r_{uv} \in R_{ij}}N_{uv}}} = {0\mspace{14mu}{or}\mspace{14mu} 1}}};}$$\mspace{79mu}{{{{and}\mspace{14mu}{i_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)}} = 0},{{{otherwise}.};}}$$\mspace{79mu}{{wherein}\mspace{14mu}{\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}}}$is the total number of active connections using wavelength λ on link ij;γ_(ij) comprises a weight of using wavelength λ at link ij, wherein apotential function of a clique associated with link ij is γ_(ij) ifwavelength λ is used at link ij; and 1−γ_(ij) if wavelength λ is notused at link ij, wherein Z_(N) is a normalization constant.
 10. Themethod of claim 9, wherein the cross layer model comprises the followingequation:P(X _(f) _(sd) ,S _(f) _(sd) ,N|R _(f) =f _(sd))=P(S _(f) _(sd) ,X _(f)_(sd) |N,R _(f) =f _(sd))P(N|R _(f) =f _(sd)), where X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ) and S_(f) _(sd) =(S_(i):V_(i)∈V_(f) _(sd) ),P(S_(f) _(sd) ,X_(f) _(sd) |N,R_(f)=f_(sd)) corresponds to the physicallayer model, which characterizes the probability distribution of thestatus and the number of affected channels at each node on theattacker's route given the status of each route and the source ofattack, P(N|R_(f)=f_(sd)) corresponds to the network layer model, whichcharacterizes the probability distribution of the status of each networkroute given the source of attack.
 11. A system, comprising: a processor;and memory including executable instructions that, when implemented bythe processor: provide a cross layer model of disturbance propagation ofa disturbance in an optical network by combining a physical layer modelof the optical network and a network layer model of the optical network,the physical layer model based on the disturbance propagation having athreshold effect only on nodes along a route followed by thedisturbance; provide the physical layer model of the optical network,wherein the physical layer model comprises the following equation:${{P\left( {X_{f_{sd}},{{S_{f_{sd}}\text{|}N} = n},{R_{f} = f_{sd}}} \right)} = {\prod\limits_{i = 1}^{k - 1}\;{{P\left( {{X_{i + 1}\text{|}X_{i}},{R_{f} = f_{sd}}} \right)}{\prod\limits_{i = 1}^{k}\;{P\left( {{S_{i}\text{|}X_{i}},{N = n},{R_{f} = f_{sd}}} \right)}}}}},$where S_(f) _(sd) =(S_(i):V_(i)∈V_(f) _(sd) ), and X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ), wherein the physical layer model includesthe probability of the number of channels affected S at each networknode V given the status X of network routes n and the source of attackR_(f), where R_(f) denotes the unidirectional flow f_(sd) where theattack originates; and determine resilience of the optical network basedon the cross layer mode.
 12. The system of claim 11, wherein theexecutable instructions when implemented by the processor, provide thenetwork layer model of the optical network, the network layer modelcomprising the following equation:${{P(N)} = {\frac{1}{Z_{N}}{\prod\limits_{({V_{i}\sim V_{j}})}\;{\gamma_{ij}{\sum\limits_{r_{uv} \in R_{ij}}\;{{N_{uv}\left( {1 - \gamma_{ij}} \right)}^{({1 - {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}}})}{l_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}} \right)}}}}}}},\mspace{79mu}{{{{where}\mspace{14mu}{l_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}} \right)}} = {{1\mspace{14mu}{if}\mspace{14mu}{\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}}} = {0\mspace{14mu}{or}\mspace{14mu} 1}}};\;{and}}$$\mspace{79mu}{{{l_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}} \right)} = 0},}$otherwise, wherein $\sum\limits_{r_{uv} \in R_{ij}}\; N_{uv}$ is thetotal number of active connections using wavelength λ on link ij; γ_(ij)comprises a weight of using wavelength λ at link ij, wherein a potentialfunction of a clique associated with link ij is γ_(ij) if wavelength λis used at link ij; and 1−γ_(ij) if wavelength λ is not used at link ij,wherein Z_(N) is a normalization constant.
 13. The system of claim 12,wherein the executable instructions, when implemented by the processor,provide a cross-layer model, wherein the cross layer model comprises thefollowing equation:P(X _(f) _(sd) ,S _(f) _(sd) ,N|R _(f) =f _(sd))=P(S _(f) _(sd) ,X _(f)_(sd) |N,R _(f) =f _(sd))P(N|R _(f) =f _(sd)), where X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ) and S_(f) _(sd) =(S_(i):V_(i)∈V_(f) _(sd) ),P(S_(f) _(sd) ,X_(f) _(sd) |N,R_(f)=f_(sd)) corresponds to the physicallayer model, which characterizes the probability distribution of thestatus and the number of affected channels at each node on theattacker's route given the status of each route and the source ofattack, P(N|R_(f)=f_(sd)) corresponds to the network layer model, whichcharacterizes the probability distribution of the status of each networkroute given the source of attack.
 14. The system of claim 11, whereinthe resilience of the optical network includes an upper bound and alower bound.
 15. The system of claim 11, wherein the executableinstructions, when implemented by the processor, represent the crosslayer model graphically as a factor graph representation.
 16. The systemof claim 11, wherein the disturbance comprises crosstalk.
 17. The systemof claim 11, wherein the optical network is an all optical network. 18.A non-transitory, tangible computer readable medium having a programstored thereon for execution by an instruction execution system to modelnetwork layer and physical layer interactions and dependencies in anoptical network, the program for performing the steps of: for a physicallayer model, modeling propagation of a disturbance in an optical networkunder static network conditions based on the disturbance propagationhaving a threshold effect only on nodes along a route followed by thedisturbance, where the physical layer model comprises the followingequation:${{P\left( {X_{f_{sd}},{{S_{f_{sd}}❘N} = n},{R_{f} = f_{sd}}} \right)} = {\prod\limits_{i = 1}^{k - 1}\;{{P\left( {{X_{i + 1}❘X_{i}},{R_{f} = f_{sd}}} \right)}{\prod\limits_{i = 1}^{k}\;{P\left( {{S_{i}❘X_{i}},{N = n},{R_{f} = f_{sd}}} \right)}}}}},$where S_(f) _(sd) =(S_(i):V_(i)∈V_(f) _(sd) ), and X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ), wherein the physical layer model includesthe probability of the number of channels affected S at each networknode V given the status X of network routes n and the source of attackR_(f), where R_(f) denotes the unidirectional flow f_(sd) where theattack originates; for a network layer model, modeling a status of eachnetwork route in the optical network based on the disturbance; andcombining the physical layer model and the network layer model toprovide a cross layer model that characterizes the disturbancepropagation based on network layer and physical layer dependencies andinteractions in the optical network.
 19. The non-transitory, tangiblecomputer readable medium of claim 18, wherein the network layer modelcomprises the following equation:${{P(N)} = {\frac{1}{Z_{N}}{\prod\limits_{({V_{i} - V_{i}})}\;{\gamma_{ij}{\sum\limits_{r_{uv}\; \in \; R_{ij}}{{N_{uv}\left( {1 - \gamma_{ij}} \right)}^{({1 - \;{\sum\limits_{r_{uv}\;\varepsilon\; R_{ij}}\; N_{uv}}})}{I_{1}\left( {\sum\limits_{r_{uv}\varepsilon\; R_{ij}}N_{uv}} \right)}}}}}}},{where}$${{I_{1}\left( {\sum\limits_{r_{uv} \in \; R_{ij}}N_{uv}} \right)} = {{1\mspace{20mu}{if}{\sum\limits_{{r_{uv} \in \; R_{ij}}\;}N_{uv}}} = {0\mspace{14mu}{or}\mspace{14mu} 1}}};\mspace{14mu}{{{and}\mspace{14mu}{I_{1}\left( {\sum\limits_{r_{uv} \in R_{ij}}N_{uv}} \right)}} = 0}$otherwise, wherein $\sum\limits_{r_{uv} \in R_{ij}}N_{uv}$ is the totalnumber of active connections using wavelength λ on link ij; γ_(ij)comprises a weight of using wavelength λ at link ij, wherein a potentialfunction of a clique associated with link ij is γ_(ij) if wavelength λis used at link ij; and 1−γ_(ij) if wavelength λ is not used at link ij,wherein Z_(N) is a normalization constant; and the cross layer modelcomprises the following equation:P(X _(f) _(sd) ,S _(f) _(sd) ,N|R _(f) =f _(sd))=P(S _(f) _(sd) ,X _(f)_(sd) |N,R _(f) =f _(sd))P(N|R _(f) =f _(sd)), where X_(f) _(sd)=(X_(i):V_(i)∈V_(f) _(sd) ) and S_(f) _(sd) =(S_(i):V_(i)∈V_(f) _(sd) ),P(S_(f) _(sd) ,X_(f) _(sd) |N,R_(f)=f_(sd)) corresponds to the physicallayer model, which characterizes the probability distribution of thestatus and the number of affected channels at each node on theattacker's route given the status of each route and the source ofattack, P(N|R_(f)=f_(sd)) corresponds to the network layer model, whichcharacterizes the probability distribution of the status of each networkroute given the source of attack.